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Doctoral Dissertations Student Theses and Dissertations
1972
Structure of zero divisors, and other algebraic structures, in higher dimensional real Cayley-Dickson algebras
Harmon Caril Brown
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Recommended Citation Brown, Harmon Caril, "Structure of zero divisors, and other algebraic structures, in higher dimensional real Cayley-Dickson algebras" (1972). Doctoral Dissertations. 2088. https://scholarsmine.mst.edu/doctoral_dissertations/2088
This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. STRUCTURE OF ZERO DIVISORS, AND OTHER ALGEBRAIC STRUCTURES, IN HIGHER DIMENSIONAL REAL CAYLEY-DICKSON ALGEBRAS
by HARMON CARIL BROWN, 1941-
A DISSERTATION Presented to the Faculty of the Graduate School of the
UNIVERSITY OF MISSOURI-ROLLA
In Partial Fulfillment of the Requirements for the Degree
DOCTOR OF PHILOSOPHY in T2756 MATHEMATICS 88 pages 1972 c. I
.. ii
ABSTRACT
n Real Cayley-Dickson algebras are a class of 2 - dimensional real algebras containing the real numbers, com- plex numbers, quaternions, and the octonions (Cayley numbers) as special cases. Each real Cayley-Dickson algebra of dimension greater than eight (a higher dimen sional real Cayley-Dickson algebra) is a real normed algebra containing a multiplicative identity and an inverse for each nonzero element. In addition, each element a in the algebra has defined for it a conjugate element -a analogous to the conjugate in the complex numbers. These algebras are not alternative, but are flexible and satisfy the noncommutative Jordan identity. Each element in these algebras can be written A= a 1+ea2 where e is a basis element and a 1 ,a2 are elements of the Cayley-Dickson algebra of next lower dimension.
Results include the facts that for each real Cayley Dickson algebra aiaj = ai+j and (aib)aj = ai(baj) for all integers i,j and any a,b in the algebra. The major result concerns zero divisors.
MAJOR THEOREM. Let A= a 1+ea2 , B = b 1+eb2 , A,B ~ 0 be elements of any real higher dimensional Cayley-Dickson algebra. Let a 1 and a 2 not be divisors of ~· Then AB = 0 if and only if
1) )a1 ,b1 ,a2 ( = -(a1 ,a1 ,b2 )+(a2 ,a2 ,b2 )+IN(a2 )-N(a1 )Jb2 ,
2) (a1 ,b1 ,a2 ) = (a1 ,a1 ,b2 )+(a2 ,a2 ,b2 )+[N(a2 )+N{a1 )Jb2 , iii
where )A,B,C( = (AB)C+A(BC), (A,B,C) = (AB)C-A(BC), and
N (a) = aa. iv
ACKNOWLEDGEMENT
The author expresses his appreciation to his major advisor, Dr. A. J. Penico, for his help in the selection of this dissertation topic and for his assistance and encour agement in preparation of this dissertation.
The author is indebted to Dr. Peter Sawtelle for his helpful comments in the writing of this manuscript.
The author is also indebted to Deidra Noland for typing this manuscript under somewhat difficult circumstances.
Deepest gratitude is extended to my wife and family for their patience, understanding, and encouragement during these years of study. v
TABLE OF CONTENTS
Page ABSTRACT • . • • ...... ii ACKNOWLEDGEMENT. . . iv
I. INTRODUCTION. • ...... • 1 II. REVIEW OF LITERATURE. • ...... • 3 III. BASIC CONCEPTS. . 7 A. Definitions •• ...... 7 B. Basic Consequences of the Definitions • • • 9
IV. THE CAYLEY-DICKSON ALGEBRAS . .17 v. EXPONENT PROPERTIES . • • • ...... • .24 VI. BASIS ELEMENT PROPERTIES •• • 30
VII. ZERO DIVISORS .43
VIII. EXAMPLES AND COUNTER EXAMPLES • .60 IX. CONCLUSIONS • ...... 70 BIBLIOGRAPHY .71 VITA ••• . . . .74 APPENDICES ...... 75 A. Basis Multiplication Table for Octonions •••• 75
B. Basis Multiplication Table for A4 • • . • .76 C. PL-1 Program for Character String Multiplication of Cayley-Dickson Algebra Elements .••••••.•••.•••• 78 1
I. Introduction.
This paper is a study of the properties of higher dimensional real Cayley-Dickson algebras. The real Cayley
Dickson algebras are an infinite class of 2n-dimensional algebras over the real field. They include all alternative finite dimensional division algebras over the reals: The real numbers, complex numbers, quaternions, and the octonions. The quaternions are a four dimensional noncom mutative real algebra devised by w. Hamilton. The octonions (usually called the Cayley numbers) are an eight dimensional noncomrnutative and nonassociative real algebra devised by
A. Cayley. L. E. Dickson in 1919 [11] devised the process whereby each of these algebras generates the algebra of next larger dimension. This class of algebras is thus called the Cayley-Dickson algebras.
Cayley-Dickson algebras up to dimension eight over arbitrary fields (and particularly over the real field) have been extensively studied. It appears that Cayley-Dickson algebras of dimension higher than eight (higher dimensional
Cayley-Dickson algebras) have been studied relatively little. There are several reasons why a study of higher dimen sional real Cayley-Dickson algebras could be valuable.
Each such algebra is a normed algebra over the reals with the added property that each nonzero element has an inverse.
Moreover, the elements of each such algebra satisfy the 2 conditions necessary to permit their use in relativistic quantum mechanics, yielding perhaps, more insight into quantum mechanical phenomena than is presently available.
Lastly, such a study could help classify types of algebras in the large class of nonassociative algebras, the noncom mutative Jordan algebras.
This paper examines, in particular: The algebraic properties of the zero divisors of higher dimensional real
Cayley-Dickson algebras; certain identities which hold (or fail to hold) in these algebras; and the properties of negative integer exponents in these nonassociative algebras.
We indicate the end of a proof in this paper by the symbol//, after the notation of K. G. Kurosh [20]. 3
II. Review of the Literature.
L. E. Dickson Ill] in the Annals of Math in 1919, devised a scheme whereby an infinite class of algebras of dimension 2n could be constructed which contains the real and complex numbers, the quaternions, and the octonions. These algebras are called Cayley-Dickson algebras in honor of Dickson and Arthur Cayley who developed the octonions. Dickson was writing about the "Eight Square Problem", the solution to which was given by the following theorem due to A. Hurwitz in 1898. 2 The identity (xl + ... + xn 2) (yl2 + . . . + Yn 2) = 2 + + z 2) where the z . are linear in x. (zl . . . n l. l. and y. can hold only for n = 1,2,4,8. l. For examples of this identity, for each specified n, Dickson used the real numbers, the complex numbers, the quaternions, and the octonions. One of the consequences of Hurwitz's theorem is that the only Cayley-Dickson algebras in which the norm of a product equals the product of the norms are of dimension 1,2,4,8. An excellent account of the search for division algebras over the real field and the relation of that search to the eight square problem is given by Charles Curtis [10]. This vein of work was completed in 1958 by R.Bott and J. Milnor [8], [23]. They proved that the only division algebras over the reals are of dimension 1,2,4 and 8. Hence the first four Cayley-Dickson algebras over the real numbers 4
are of primary importance. A. A. Albert examined Hurwitz's proof and Dickson's process in 1941, and generalized these ideas to the idea of quadratic forms Il]. In 1946 I2] he considered the idea of an absolute-valued algebra, and showed that all real algebras are normed algebras. In 1948 [3] he examined the property of power associativity in rings as well as the properties of flexibility and trace-admissibility. Each Cayley-Dickson algebra enjoys these properties. In 1948 [4] he studied trace-admissible algebras and displayed several more properties of the trace operator which hold in any Cayley-Dickson algebra. It was R. D. Schafer in 1954 [26], however, who came back to examine more closely the Cayley-Dickson algebras themselves. Indeed, the chief references for the subject are his 1954 paper and his 1966 book An Introduction to Nonassociative Algebras [28]. In his paper, he derives certain elementary properties of these algebras and examines chiefly their derivation algebras. He shows, for example, that all Cayley-Dickson algebras are flexible. He also shows that the basis elements of all such algebras are alternative, even though the algebras are not alternative if the dimension is greater than eight. From here, the investigation seems to follow two widely separating paths. One path of study is the investigation and classification of the nonassociative algebras in general. In 1955 [27], Schafer was able to classify simple 5 noncommutative Jordan algebras of characteristic 0 into three classes: simple (commutative) Jordan algebras, simple quasiassociative algebras, and simple flexible algebras of degree two. He then commented that much remains to be learned about that last classification inasmuch as it contains the Cayley-Dickson algebras for which relatively little is known. Most of the work in these areas of late, has been in trying to find identities characterizing cer tain classes of nonassociative algebras. These have in cluded the concepts of standard algebras, generalized standard algebras, accessible algebras, generalized acces sible algebras, and algebras with alternativity conditions. These have been studied by such people as A. J. Penico [24], R. D. Schafer [29], [30], R. E. Block [7], E. Kleinfeld [18], M. H. Kleinfeld, J. F. Kosier [19], and K. McCrimmon [21]. The other path has been to study the quaternions and octonions extensively with some of the techniques of other areas of mathematics. In particular, the ideas of functional analysis have been applied to them by H. H. Goldstine and L. P. Horwitz [14], [15], [16], and more recently by J. Jamison [17]. Number theory and basic algebra methods have been applied to the quaternions and octonions by s. Eilenberg and I. Niven [12], M. J. Wonenburger [32], and H. S. Coxeter [9]. The octonions and their relation to the Dirac wave equation in physics have been studied by R. Penney [25]. 6
The last paper to date, to the author 1 s knowledge, specifically mentioning the Cayley-Dickson algebras is R. D.
Schafer's paper in 1970 concerning forms permitting composi tion [31]. In this paper, he does not mention any signifi cant new results about the Cayley-Dickson algebras. 7
III. Basic Concepts.
A. Definitions.
The following ideas, which can be found in [21] and in
[28], will be used frequently in our study.
Definition 3.1. A ring R is an additive abelian group with a multiplication satisfying the distributive laws
(x + y)z = xz + yz and z(x + y) = zx + zy for any x,y,z in
R. (Note that (xy)z = x(yz) is not assumed.)
Definition 3.2. An algebra A is a ring which is also a vector space over a field F with a bilinear scalar multipli
cation satisfying the scalar associative law: r(xy) = (rx)y = x(ry), for all r in F and for all x,y in A. Generally, what we have defined is called a nonassocia
tive algebra to emphasize that the associative law is not
assumed. We will restrict our attention to algebras with a multiplicative identity element over the real field. It is
important to note that the real field is of characteristic
zero. In the following, we consider algebras rather than
rings; but obviously, many definitions and results do not
depend on scalars and hence are true for rings as well.
Definition 3.3. An element a of an algebra A is a
zero divisor if a ~ 0 and there exists some element b ~ 0 of A for which ab = 0 or ba = 0. The elements a and b will be referred to as mutual zero divisors if they are zero divisors and ab = 0 or ba = 0. Definition 3.4. A division algebra is an algebra in 8 which every element has an inverse and there are no divisors of zero.
Definition 3.5. The commutator [x,y] is given by [x,y] = xy - yx.
Definition 3.6. The associator (x,y,z) is given by
( X 1 y 1 Z ) = ( XY ) Z - X ( y Z ) •
Definition 3.7. The nucleus of an algebra is the set of all elements x of an algebra A for which (x,y,z) =
(y,x,z) = (y,z,x) = 0 for all y,z in A.
Definition 3.8. The center of an algebra is the set of all elements in the nucleus of the algebra which also commute with all elements of the algebra.
Definition 3.9. An involution (involutorial anti- automorphism) is a linear operator x ~ x such that xy = yx and x = x. We call x the conjugate of x.
In this paper, ~will only b~.concerned with involu tions for which x + x and xx are in the center of the algebra. Dr. A. J. Penico has suggested that these might best be called "centered involutions".
Definition 3.10. The trace T (x) of the element X is defined by T(x) = X + x.
Definition 3.11. The norm N(x) of the element X is defined by N (x) = XX = XX. Definition 3.12. The flexible property is (xy)x = x{yx). If the flexible property is satisfied for allele- ments of an algebra, the algebra is said to be flexible.
Definition 3.13. The right alternative property is 9
(yx) x = yx 2 , and the left alternative property is x(xy) = 2 X Y• If both properties are satisfied by all elements of an algebra, the algebra is said to be alternative.
B. Basic consequences of the definitions.
In this section, we observe basic properties of the terms defined previously so that efficient algebraic manipu- lations will be available to examine the Cayley-Dickson algebras. Many of these observations are well known, and are included to aid the reader. The commutator is defined by [x,y] = xy - yx. If
[x,y] = 01 then xy = yx. Clearly, [x,x] = 0 and [x,y] = -[y,x]. In an algebra, a scalar r may be thought of as r • 1 where 1 is the identity element of the algebra, so we have [r,x] = 0 for all x in the algebra. Moreover, r[x,y] = [rx,y] = [x,ry]. Commutators are also additive in each argument, i.e. , [ x + y, z] = [ x, z] + [ y, z] and [ x, y + z] =
[x,y] + [x,z]. Thus commutators are linear in each argu- ment. Since the trace is in the center of an algebra, we have the following additional properties of the commutator and the conjugate. [x,y] = -[x,y] = -[x,y] = [x,y] = -[x,y]. These equalities follow from [x,y] = [T(x} - x,y] = [T(x),y] +
[-x,y] = 0 - [x,y]; [x,y] = [x,T (y) - y] = [x,T(y)] +
[x,-y] = -[x,y]; and [x,y] = xy- yx = xy- yx = yx- xy =
[y,xl = [y,xl = -[x,y]. 10
The associator is defined by (x,y,z) = (xy)z- x(yz): (x,y,z) = 0 if and only if (xy)z = x(yz). Moreover, if r is an element of the center, then (r,x,y) = (x,r,y) = (x,y,r) = 0. The associator is also additive in each argument, i.e., for example, (x + y, z ,w) = (x, z ,w) + (y, z ,w). Thus the associator is linear in each argument. If an algebra A is flexible, then we have (xy)x = x(yx); or, in terms of associators, (x,y,x) = 0 for all x,y in A. By the additive property this means 0 =
(x + w,y,x + w) = (x,ylx) + (x,y,w) + (w 1 y,x) + (w,y 1 w) =
(x,y,w) + (w,y 1 x). Thus in a flexible algebra (x 1 y,w) = -(w,y,x), for all x,y,w in A. The following relations between conjugates and asso ciators hold in algebras with involution.
(xI y I z) = - (x, y, z > = - (x, y, z > = - (x, y, z >
= ex, y, z > = ex, y I z> = ex, y, z> = -
= (z,y,x) These properties follow from the observation that
(x,y,z) = (T(x) - x 1 y,z) = (T(x),y,z) + (-x,y,z) = 0- (x,y,z); and (z,y,x) = (zy)x- z(yx) = x(yz) - (xy)z =
- (x,y,z) = cx,y,z>. The "Four Identity" is: (xy,z,w)- (x,yzlw) +
(x,y,zw) = x(y,z,w) + (x,y,z)w. This is easily verified as being true in any ring, and will be useful in certain calculations. 11
Just as the flexible property can be expressed in
terms of associators, the alternative properties can be as well. The right alternative property (yx)x = yx2 can be written as (y,x,x) = 0, and the left alternative property
x(xy) = x 2y can be written as (x,x,y) = 0. If an algebra
is alternative, i.e., both the left and right alternative properties hold, then 0 = (x + y,x + y,z) = (x,x,y) +
(x,y,z} + (y,x,z) + (y,y,z) = (x,y,z) + (y,x,z). Thus in
an alternative algebra (x,y,z) = -(y,x,z}. Similarly, in an alternative algebra (x,y,z) = -(x,z,y). Moreover, any
two of (x,x,y} = 0, (y,x,x) = 0, or (x,y,x} = 0 imply the
third. For example, if the alternative laws hold, then
0 = (x,x + y,x + y) = (x,x,x) + (x,y,x) + (x,x,y) + (x,y,y)
= (x,y,x).
The trace T(x) has been defined to be in the center of the algebra. Thus if r = r, then r is in the center since T(r) = r + r = 2r. For such an r we have T(rx) = rT(x),
since rx + rx = rx + xr = rx + rx = r(x + x}. It is also
clear from the definition that T(x} = T(x).
THEOREM 3.1. In any flexible algebra with involution, T(xy) = T(yx} and T([xy]z) = T(x[yz]).
Proof. Since [x,y] = [x,y], we have xy- yx = xy- yx or xy + yx = xy + yx. Therefore, xy + xy = yx + yx and so T(xy) = T(yx). Also since (x,y,z) = -(x,y,z), we have
(x,y,z) + (x,y,z) = 0, i.e., T((x,y,z)) = 0. Thus
T([xy]z- x[yz]} = T([xy]z) - T(x[yz]) = 0.// 12
We also note that T(xy) = T(yx) since xy- + xy- = xy + yx = yx + yx. Hence T(xy} = T(xy) because T(xy) = T(yx}.
It is not true, however, that T(xy} = T(xy}, as can be seen from the complex numbers with the usual conjugate. There, trace corresponds to twice the real part of each complex number. Thus T([l + i]i) = -2, but T([l- i]i) = +2. Finally, we observe a relationship that exists between T(xy) and T(xy).
LE~1A 3.2. T(x)T(y) = T(xy) + T(xy). Proof. T(x)T(y} = (x + x} (y + y} = xy + xy + xy + xy = (xy + yx} + {xy + xy + xy- yx) = T(xy) + (xy + x[y + y] - yx} = T(xy} + (xy + T(y}x- yx) = T(xy) + (xy + [T(y) - y]x} = T(xy) + (xy + yx} = T(xy} + T(xy}.// In an algebra with involution, several useful rela- tionships exist between the conjugate and the operation of association. The following will illustrate.
THEOREM 3.3. In an algebra with involution, the following hold: i) x(xy) = x(xy) ii) {yx)x = (yx)x -2 -2 iii) x{x y} = x (xy}. If the algebra is also flexible, then: iv) x{xy) = (yx)x v) x 2 (yx} = (x2y)x. Proof. We use the properties of trace to make the calculations: 13
i ) X ( xy ) = ( T (X ) - X ) ( xy ) = T (X ) ( xy ) - X ( xy ) = x[T(x)y- xy] = x([T(x) - x]y) = x(xy).
ii) is shown in a similar manner. For iii), consider, x(x2y) = x[x(T{x)- x)y] = x[T(x)x- N(x))y] = x[T(x) (xy) - N(x)y] = T(x) [x(xy)] - N(x) (xy). Now by i) this equals T(x) [x(xy)] - N(x) (xy) = [T(x)x- N(x)] {xy) =
[ N (x) + x 2 - N ( x) ] ( xy ) = x 2 ( xy ) •
Now assuming the flexible property, iv) follows from x(xy) = -(x,x,y) + (xx)y = -(x,x,y) + N(x)y = (y,x,x) +
N(x)y = (yx)x- yN(x) + N(x)y = (yx)x.
To show v) holds, recall that an associator is zero if 2 any entry is in the center. Then (x ,y,x) =
(x[T(x)- x],y,x) = (T(x)x- N(x),y,x) = (T{x)x,y,x) -
(N(x),y,x) = T(x) (x,y,x) - 0 = 0.//
Definition 3.14. A noncommutative Jordan algebra is a noncommutative flexible algebra satisfying (x2 ,y,x) = 0.
COROLLARY 3.4. Any flexible noncommutative algebra with involution is a noncommutative Jordan algebra.
Proof. This follows from the proof of v) above.//
The following is given by Schafer [28] as an exercise. 2 LEMMA 3.5. In any flexible algebra: (x ,y,x) = 2 2 2 2 x(x y) - x (xy) = (yx )x - (yx)x .
Proof. x(x2y) - x 2 (xy) = x[ (xx)y] - (xx) (xy) =
-(x,x,xy) + x[x(xy)] + x[ (x,x,y) - x(xy)] = (xy,x,x) - x(y,x,x) by the flexible property. Now, this last expres sion becomes [(xy)x]x- (xy)x2 - x[ (yx)x] + x(yx2 ) =
-(xy)x2 + x(yx2 ) - x[(yx)x] + [(xy)x]x = -(x,y,x2 ) - 14
2 2 x[(yx)x] + [x(yx)]x=-(x,y,x) + (x,yx,x) = (x ,y,x).
Likewise (yx2 )x- (yx)x2 = [y(xx)]x- (yx) (xx) =
(yx,x,x) - [ (yx)x]x + [ (yx)x - (y,x,x) ]x = (x,x,y)x -
(x,x,xy) by the flexible property. This then becomes 2 2 2 (x y)x- [x(xy)]x- x (xy) + x[x(xy)] = (x ,y,x) + 2 2 x[x(xy)]- x[(xy)x] = (x ,y,x)- x(x,y,x) = (x ,y,x).//
LEMMA 3.6. In any noncommutative Jordan algebra we have (x,x,yx) = (x,x,y)x and (xy,x,x) = x(y,x,x).
Proof. We use the "Four Identity" which is valid in any algebra.
(xx,y,x)- (x,xy,x) + (x,x,yx) = x(x,y,x} + (x,x,y)x.
Now by the noncommutative Jordan identity and the flexible property, we have (x,x,yx} = (x,x,y)x. 2 Similarly, (xy,x,x) - (x,yx,x) + (x,y,x ) = x(y,x,x) +
(x,y,x)x. Thus, (xy,x,x) = x(y,x,x).//
COROLLARY 3.7. In any noncommutative Jordan algebra we have [x (xy) ]x = x[x (yx)] and [ (xy)x]x = x[ (yx)x].
Proof. 0 = (x,x,yx) - (x,x,y)x = x 2 (yx) - x[x (yx)] - 2 2 (x y)x + [x(xy)]x = -(x ,y,x) + [x(xy)]x- x[x(yx)] =
[x(xy)]x- x[x(yx)]. The other part is similar.//
We have defined the norm of x to be N(x) = xx, and have specified that N(x) is in the center. The presence of norm and trace lead to the next lemma.
LEMMA 3.7. Every element in an algebra with involu tion satisfies the quadratic equation: x 2 - T(x)x + N(x)l = o. 15
Proof. x 2 - T(x)x + N(x)l = x2 - (x + x)x + XX= 0.//
LEMMA 3.8. The following hold in any algebra with in volution: 2 i) N(r) = r , if r = r
ii) N(rx) = r 2N(x), if r = r
iii) N(x) = N(x). Proof. Each is clear from the definition of norm.// THEOREM 3.9. In any flexible algebra with involution the following hold:
i) N(xy) = N(xy)
ii) N
iii) N(xy) = N (yx). Proof. The proof relies on the fact that the trace is in the center of the algebra.
i) N (xy) = (xy) (yx) = [x (T (y) - y) 1 (yx) =
[T(y)x- xy] (yx) = [T(y)x] (yx) - (xy) (yx) = [T(y)x] (yx) -
(xy) r (T(y) - Y.>xl = [T (y)xl (yx) - T (y) r
LEMMA 3.10. In any algebra with involution we have:
N(x ± y) = N(x) + N(y) ± T(xy). 16
Proof. N(x + y) (x y) (x - - - = + + y) = (x +- y) IV. The Cayley-Dickson Algebras. R. D. Schafer's book 128] gives a clear description of the process that Dickson devised to generate the Cayley Dickson algebras. Let A be an algebra with identity 1 and with an involution. Let A have dimension n. Then we con- struct an algebra B of dimension 2n over the same field as A and having A isomorphic to a subalgebra of B. Let B con sist of all ordered pairs (a1 ,a2 ) where a 1 and a 2 are ele ments of A. Let addition and multiplication by scalars be defined componentwise. Moreover define a multiplication in B by (a1 ,a2 ) (a3 ,a4 ) = (a1 a 3 + va4a 2 ,a1 a 4 + a 3a 2 ), where vis some nonzero scalar. Defining (1,0), the identity element of B, as 1, then: A' = { (a,O) I a in A} is a subalgebra iso morphic to A; e = (0,1) is an element of B such that e 2 = vl; and B is the vector space direct sum B =A'$ A'. For our purposes, we restrict our field to be the real numbers, and v to be -1. It will be handier to consider elements of B to be of the form x = a 1 + ea2 with multipli cation given by (a1 + ea2 ) (a3 + ea4 ) = (a1a 3 - a 4a 2 ) + e(a1a 4 + a 3a 2 ). This notation will be consistently used. Define X by X = a 1 - ea2 if X = a 1 + ea2 • Then it is easy to see that xy = yx since a ~ a is an involution for A. Thus x ~ x is an involution for B. Trace and norm are de- fined as before with the observation that for x = a 1 + ea2 , T(x) = T(a1 ) and N(x) = xx = (a1 + ea2 > (a1 - ea2 ) = (a1a 1 + a 2a 2 ) + e(-a1a 2 + a 1a 2 ) = N(a1 ) + N(a2 ). In any 18 Cayley-Dickson algebra, the inverse of any nonzero element exists and is defined by x-l = x/N(x). Then xx-l = x[x/N{x)] = xx/N{x) = 1. Since this is the case, there are no non-trivial proper ideals in any of these algebras. The reader should take note that all of the material presented above is used repeatedly in the remainder of this paper. If A is the real numbers, then the B given by the Cayley-Dickson process is the complex numbers, e is repre- sented by i, and the involution is the familiar conjugate. 2 2 Moreover, T(x) = 2 Re(x) and N{x} = a 1 + a 2 where x = a 1 + ia2 • We note that in going from the real to the com plex field, order is lost, i.e., the complex numbers are not ordered. If A is the complex numbers, B will be the quaternions, e will usually be represented by j, and ij = k. In going from the complex numbers to the quaternions, commutativity is lost since ji = -k. If A is the quaternions, B will be the octonions (usually called the Cayley numbers}. The octonions are not associative, but are alternative and are a division algebra. If A is the octonions, the 24-dimensional algebra ob- tained is the smallest Cayley-Dickson algebra which is not a division algebra since it has zero divisors. It is not alternative for the same reason. It has no name, and will be referred to in this paper as A4 in keeping with the notation of Schafer [26]. To the author's knowledge, A4 19 has not been studied extensively, and very little is known about its properties. This paper is an attempt to answer some questions about A4 ; in particular, questions regarding zero divisors. In our investigation, however, many other facts were discovered which are true for all Cayley- Dickson algebras. There is an alternate way of looking at each of these algebras. On can consider the complex numbers as the vector space generated by {l,i} over the real numbers. Likewise the quaternions can be thought of as the vector space over the real numbers having basis elements {l,i,j,k} with multiplication of basis elements defined by the following table: 1 i j k 1 1 i j k i i -1 k -j j j -k -1 i k k j -i -1 An artibrary element thus has the form r 0 + r 1i + r 2 j + r 3k, where the r's are real numbers. Similarly one can view each of the Cayley-Dickson algebras as a vector space over the reals. In order to let the basis of A be denoted facilitate notation, n 2 2 {l=e0 ,e1 , ••• ,e2n_ 1 }, where e 0 = e 0 and ei = -e0 for i ~ 0. Thus an element of the algebra can be represented the linear combination x = L1.· r.e., i = 0, ..• ,2n-l. as 1. 1. 20 n With this representation X r. e., i 0, ••• ,2 -1. = roeo - L·1 1 1 = 2 n Also T(x) 2r , and N(x) r. I i 0, ••• ,2 -1. Note = 0 = L·1 1 = also that but -e. -e. if i 0. In the remainder eo = eo, 1 = 1 ~ of this paper, we shall refer to the subspace re0 as simply .. the reals r" with the meaning being clear from context. LEMMA 4.1. N(x) = 0 if and only if x = 0. Proof. N(x) 0 if and only if r. = 0 for all i. = 1 This is equivalent to saying x = 0.// Tables for the basis element multiplication for A3 , and for A4 will be found in Appendix A and Appendix B. It would seem that as the Cayley-Dickson process continues, some algebraic property would be lost at each step just as in the first three steps. This paper will indicate several ways that A5 differs from A4 , but what is lost in higher dimensions remains largely unknown. Another viewpoint is to consider what properties, if any, are enjoyed by all Cayley-Dickson algebras. This last question is easily answered at least in part. For example, if a new norm is defined by n(x) = IN(x), then each Cayley-Dickson algebra becomes a normed linear space over the real numbers with the added property of having a multiplication and an inverse for each nonzero element. This explains why the techniques of functional analysis have been useful in the past in studying the quaternions and the octonions. Since it has been known for many years that the octonions are the only alternative, nonassociative division 21 algebra over the real numbers, an interesting question answered by Schafer is whether or not any higher dimensional Cayley-Dickson algebras are alternative. The answer is no, but they are all flexible as the following shows. In 1954, R. D. Schafer [26] proved the following theorem. We shall give an alternate proof here. THEOREM 4.2. Any Cayley-Dickson algebra is flexible. Proof. This is obvious for any associative algebra. It therefore suffices to show that if A is flexible, then n An+l is flexible. Let A = a 1 + ea2 and B = b 1 + eb2 be elements of An+l. Then (A,B,A) = (A,B,A) = [ (a1-ea2 ) (b1-eb2 )] (a1+ea2 ) - (a1-ea2 ) [ (b 1-eb2 > (a1+ea2 ) 1 = [ (a1£1-b2a 2 > +e (-a1 b 2 -51 a 2 ) 1 (a1 +ea2 > + (-a1+ea2 ) [ (b1a 1+a25 2 )+e(b1a 2-a1b 2 )] = £ (al5l>al- (b1a 2 )a2+(a1b 2 )a2 J + e[(b1a 1 >a2- [(a1 ,£1 ,a1 ) + T(a2 (b2a 1 ))- T(al (a2b 2 )) + a 2 (a2b 1 ) (b1a 2 )a2 ] + e[ (b1a 1 + b1a 1 )a2 - a 1 c£1a 2+b1a 2 )J = [(a1 ,b1 ,a1 )+T(a1 (a25 2 >>-T + e[T(b1 ) (a1a 2 ) - T(b1 ) (a1a 2 )] = (a1 ,b1 ,a1 )+[(a2 ,a2 ,b1 )- (b1 ,a2 ,a2 )] which is zero if An is flexible.// We note that xyx is now unambiguous since (xy)x = x (yx). THEOREM 4.3. Let A= a 1+ea2 and B = b 1+eb2 be elements of a Cayley-Dickson algebra of any dimension. 22 Then (A,A,B) = [ (a1 ,b2 ,a2 )+(a2 ,a2 ,b1 )+(a1 ,a1 ,b1 )l + e[(a1 ,a1 ,b2 )-(a1 ,b1 ,a2 )+(a2 ,a2 ,b2 )]. Proof. We first observe that (A,A,B) = -(A,A,B) = -N(A)B + A(AB). Now A(AB) = (a1+ea2 )[ (a1-ea2 ) (b1+eb2 )] = (a1+ea2 ) [ +e (bla2)a2] + e[al (alb2)-al(bla2)+(albl)a2+(b2a2)a2] = [-Ca1 ,b2 ,a2 >+a1 ca1b 1 )+(b1a 2 >a2 l+e[ ca 1 ,b1 ,a2 >+a1 (a1b 2 ) + (b2a2)a2]. Also -N(A)B = -(a1a 1 + a 2a 2 ) (bl + eb2 ) = [-(alal)bl~(a2a2)bl] + e[-(alal)b2-(a2a2)b2]. so (A,A,B) = [-Ca 1 ,b 2 ,a 2 )+a 1 ca 1 b 1 >~ca 1 a 1 >b 1 + (bla2)a2-(a2a2)bl] + e[(al,bl,a2)+al(alb2)-(alal)b2 + (b2a 2 )a2-(a2a 2 )b2 ]. Now since N(a) = aa = aa and is in the center, we have: (al,al,bl)+(bl,a2,a2)] + e[(al,bl,a2)-(al,al,b2) + (b2 ,a2 ,a2 )]. Now using the flexible property and the properties of conjugates and associators, we have the result.// From this we get another theorem of Schafer's (cf. [28], p. 46) as a corollary. COROLLARY 4.4. A Cayley-Dickson algebra B is alterna- tive if and only if the generating algebra A is associative. Proof. If B is alternative, our equation for (A,A,B) reduces to (a1 ,b2 ,a2 )-e(a1 ,b1 ,a2 ) which is identically zero if and only if the algebra A is associative.// 23 Finally, we comment here, that it is easy to see that the center of each Cayley-Dickson algebra is exactly Fl, the field times the identity. 24 V. Exponent Properties. It is a well known fact that each Cayley-Dickson algebra is 11 power associative." Definition 5.1. An algebra is power associative if 1 i+l . . . '+' x = x, x = xx1 , and x1xJ = x1 J for all x in the algebra, and i,j positive integers. .. Th1s. means that ( x m n,x ,xp ) = 0 for all pos1t1ve integers m,n,p and x any element of a Cayley-Dickson algebra. What we wish to do in this section is extend this definition to integer powers (non-positive as well), and show that all Cayley-Dickson algebras still obey this associative property. - n n LEID1A 5.1. In any Cayley-Dickson algebra, (x) = x • Proof. We use power associativity and induction on n. It suffices to note that: (x)k+l = (x)kx = (xk)x = x(xk) = xk+l.// Although every element in a Cayley-Dickson algebra has an inverse, it is not true that it is unique since -1 zero divisors exist. For example, if aa = 1 and if ab = 0, -1 then a(a +b) = 1 also. To avoid this problem, we adopt the convention that a-l will always mean a/N(a). Define 0 a = 1, for a 'f 0. -1 n n-1 LEMMA 5. 2. In any Cayley-Dickson algebra,a a = a for all integers n greater than or equal to 1. Proof. We use induction on n. It is clear that the statement is true for n = 1 by our definitions. For n > 2, 25 assume a-lak = ak-l for 1 < k < n-1. Recall that a satis fies the quadratic equation a 2 - T(a)a + N(a)l = 0. Then k k-1 k-2 -1 n -1 n-2 a = T(a)a - N(a)a • Thus a a = a [T(a}a-N(a}]a = [T(a)a-N(a)]a-lan-2 = [T(a)a-N(a)]an-3 = an-1 .;; COROLLARY 5.3. The only idempotent elements of An are 0 and 1. -1 2 Proof. a if and only if a = 0 or a = a a = a-1a = 1.// COROLLARY 5.4. For n > 1, we have: -1 n n -1 i) a a = a a - n ii) a a Proof. Fori) we note that (an-l,a,a-1 ) = -1 n-1 n -1 n-1 -(a ,a,a ) by flexibility. Thus a a -a = n-1 -1 n -a +a a = 0. -1 n n-1 n -1 To see ii) note that [a/N(a)]an = a a = a = a a = an[a/N(a)].// COROL~ARY 5.5. There ~no nilpotent elements in any Cayley-Dickson algebra, i.e., an= 0 implies a= 0. Proof. Suppose a ~ 0. Let n be the smallest positive integer such that an= 0. Then an-l = a-lan= a-l • 0 = 0, n-1 so a = 0, a contradiction.// THEOREM 5.6. NP(a) = N(aP), where pis a positive integer and NP(a) is [N(a)]P. k k-1 k - k-1 Proof. First note that 0 = (a ,a,a ) = (a ,a,a ) k - - k-1 =(a ,a,(a) ). proof of the theorem is by induction on p. Assume the kk statement true for 1 ~ p ~ k-1. Then N(ak) = a a = 26 (aka) (a)k-l = N(a)ak-l(a)k-l = N(a)ak-lak-l = N(a)N(ak-l) = N(a) [N(a)]k-l = Nk(a).// COROLLARY 5. 7. - m Proof. [a/N(a)] = Notation. Let a-m be defined to be (a-l)m. LEMMA 5. 8. T(a2 ) = T2 (a) - 2N(a). 2 2 ~ 2 -2 2 2 Proof. T(a ) =a +a =a +a =a +a +2aa- 2aa = Theorem 5.6 is the best result possible for norms of products; for, although N(ab) = N(a)N(b) is valid for Cayley-Dickson algebras of dimension 1,2,4, and 8, it is not valid for higher dimensions, because of zero divisors. For example, if a,b ~ 0, but ab = 0, then 0 = N(ab) ~ N(a)N(b). In fact, as an example in the next chapter shows, both N(ab) > N(a)N(b) and N(a'b') < N(a')N(b') can occur. Definition 5.2. An algebra is integer power associa- . . f l i+l i i J. "+" t lVe l X = X, x = xx , and x x = x 1 J for all x in the algebra and i,j any integers. To the author's knowledge this idea is not found in the literature. The following leads to the fact that the Cayley-Dickson algebras are integer power associative. LE~1A 5.9. For m,n positive integers, (am)-1 (an)-l = ( a m a n)-1 • Proof. = (a)m+n = Nm+n(a) 27 m - n LE!vlMA 5.10. For m,n positive integers, a (a) ::::; m n-1 m - n-1 Proof. Note that 0 = (a ,a,a ) = (a ,a,a ) = m - - n-1 m - - n-1 (a ,a, (a) ) = (ama) (a) n-1 - a [a(a) ]. Hence am(a)n = N(a)am-l(a)n-1. By induction we have am(a)n = Nj(a)am-j (a)n-j for all j ' 1 -< j ~ m,n. Now if n = m, then am{a)n = Nn(a). If n < m, then am(a)n = Nn{a)am-n for j = n. If n > m, then am(a)n = ~(a) (a)n-m = Nm (a) (a) n-rn(~=m (a)J = ~+n-rn (a) [(a) n-m/Nn-m (a) 1 = Nn m (a)7 ~(a) [a/N(a)]n-m = Nn(a) (a-l)n-m = Nn(a)am-n. The proof of the other part is similar.// rn n rn+n THEOREM 5.11. a a = a , for all integers m,n. Proof. Case 1: m or n = 0; this follows from the definition of a 0 • Case 2: m,n positive; this follows from power asso- ciativity. m n Case 3: m,n negative; a a -n -1 -ro-n -1 m+n (a-m a ) = (a ) = a • m n m -n -1 Case 4: n negative, m positive; a a = a (a ) = arn[a-n/N{a-n)] = am[(a)-n/N-n(a)] = [N-n(a)am+n]/N-n(a} = m+n a m n -m -1 n Case 5: n positive, m negative; a a = (a ) a = [ (a-m)/N(a-m)Jan = [ (a)-m/N-m(a)Jan ::::; [N-rn(a)an- (-m) ]/N-m(a) = arn+n.// THEOREM 5.12. (am)n = amn for all inte·gers rn and n. 28 Proof. Case 1: m or n = 0; the result is obvious. Case 2: m,n positive; this follows by power associa- tivity. Case 3: m,n negative; (am)n = ([(a-m)-l]-n)-1 = [ (a-m)n]-1 = (a-m)-n = a(-m) (-n) = amn. Case 4 ·. m nega t'1ve, n pos1't' 1ve; (am)n __ [(a-m)-l]n __ (a-m)-n = [[a(-m) 1n 1-l = amn. Case 5 : m pos1t1ve,· · n negat1ve;. (am)n __ [ (am)-n]-1 __ [am(-n) 1-l = amn.// We now note that it is not possible easily to extend th ese resu lt s t o f rae t 1ona. 1 exponen t s s1nce . even a 1/2 1s. ambiguous. For example, each basis element e. ~ 1 satis- 1 fies the equation x 2 = -1. An interesting side light does occur here, however, in that the square of an associator is 1n fact a negative real number, as we show below. 2 Lill"MA 5. 13. T(x) = 0 implies x = -N(x). Proof. Recall, for any x, we have x 2-T(x)x+N(x)l = 0. So, if T(x) = 0, the result follows.// 2 LID1MA 5.14. (a,b,c) = -N ( (a,b,c)). Proof. Recall that (a,b,c) = (c,b,a) = -(a,b,c) so T ( (a,b,c)) = 0. (Theorem 3.1). Thus by Lemma 5.13, the result follows.// It is now possible to show that flexibility, (a,b,a) = 0, and the noncommutative Jordan identity, (a 2 ,b,a) = 0, are special cases of a more general identity involving the associator. 29 n n LE!V'~ 5.15. (a ,b,a} = (a,b,a ) = 0 for all non- negative integers n. Proof. The statement is clear for n = 0,1,2. We use induction and the flexible property to finish the proof. n k+l Assume (a ,b,a) = 0 for all n < k. Then (a ,b,a) = (ak[T{a)-a] ,b,a) = T(a) (ak,b,a)-N(a) {ak-l,b,a) = 0.// m n LEMMA 5.16. (a ,b,a ) = 0 for all ~-negative inte- gers m,n. Proof. If m = n, the result follows from the flexible property. If m or n = 0, the result is obvious. If m ~ n, and if neither is zero, then {am,b,an) = {am-l [ T {a) -a] 1 b , an -l [ T (a) -a] ) = T 2 {a) (am-l , b , an-l ) - T{a)N{a) (am-l,b,an-2 )- T(a)N(a) (am-2 ,b,an-l)- N2 (a) {a m-2 ,b,an-2 ), which clearly vanishes by a finite induction argument. Thus the theorem follows.// m n THEOREM 5.17. (a ,b,a ) = 0 for all integers m,n. Proof. If m = n, the result is clear. If m,n are non-negative, the result follows by Lemma 5.16. If m,n are m n m n - m - n negative, then {a ,b,a ) = {a ,b,a ) = ((a) ,b, (a) ) = (Nm(a) [a-l]m,b,Nn(a) [a-l]n) = Nn+m(a) (a-m,b,a-n) = 0. m n If m is negative, and n is positive, then (a ,b,a ) = m n - m n __m -m n) -{a ,b,a) =-{{a) ,b,a) = -(N (a)a ,b,a = -~(a) (a-m,b,an) = 0. m n If m is positive, and n is negative, then (a ,b,a ) = m n - n m n -n -( a ,b,a ) = - (am,b, (a) ) = -{a ,b,N {a) a ) = -N(a) (am,b,a-n) = 0.// 30 VI. Basis Element Properties. Some very useful identities which are true in alterna- tive algebras are the Moufang identities: i) (xax)y = x[a(xy)] ii) y(xax) = [ (yx)a]x iii) (xy) (ax) = x(ya)x. Unfortunately, these identities do not hold in Cayley- Dickson algebras of dimension higher than eight. We will now show, however, that the Moufang identities and the alternative property do hold in a restricted, but useful, sense. Consider the Cayley-Dickson algebra B of dimension 2n formed by adjoining e n-l to the basis of the 2n-l 2 dimensional algebra. Call the adjoined basis element e for notational convenience. 'rHEOREM 6 . 1. (e,A,B) + (A,e,B) = 0 for all A,B in B. Proof. Let A= a 1+ea2 and B = b 1+eb2 . Then, (e,A,B) + (A,e,B) = (eA)B- e(AB) + (Ae)B- A(eB) = (-a2+ea1 ) (b1 +eb 2 ) - (O+el) [ (a1b 1-b2a 2 )+e (a1b 2+b1a 2 ) J + (-a2+ea1 ) (b1+eb 2 ) - (a1+ea2 ) (-b2+eb1 ) = [ (-a2b 1-b2a1 )+e(-a2b 2+b1a 1 )J + [ (a1b 2+b1a 2 )-e(a1b 1-b2a 2 )J + [ (-a2b 1-b2a 1 )+e(-a2b 2+b1a1 )J + [ (a1b 2+b1a 2 )+e(-a1b 1 +b 2a 2 )J = [-a2bl-b2al+alb2+bla2-a2bl-b2al+alb2+bla2J + e[-a2b2+blal-albl+b2a2-a2b2+blal-albl+b2a2] = [-T(a2 )b1 + T(a1 )b2 + T(a2 )b1 - T(a1 )b2 ] + e[-T(a2 )b2 + T(a1 )b1 - T(a1 )b1 + T(a2 )b2 ] = 0.// 31 LEMMA 6.2. The following are ----true for all A,B in B. i) (A, B, e) + (A, e, B) = 0 ii) (e,A,A) = (A,A,e) = (e,e,A) = (A, e, e) = 0 iii) (e,A,B} + (e,B,A} = 0 iv} (A, B, e) + (B,A,e) = 0. Proof. To see i) , note that (A,B,e) + (A, e, B) = - (e,B,A) - (B,e,A) = 0, using the flexible property and Theorem 6.1. For the same reasons, 0 = (e,A,A) + (A,e,A) = ( e , A , A ) ; 0 = (A , A , e ) + (A, e , A ) = (A , A , e ) ; 0 = ( e , e , A ) + ( e , A, e ) = (e , e , A) ; and 0 = (A 1 e 1 e ) + ( e 1 A 1 e ) = (A 1 e 1 e ) proving ii). To see iii), note that (e,A,B) = -(A,e,B) = (B 1 e 1 A} = - (e 1 B 1 A). Similarly for iv) 1 (A 1 B,e) = - (A 1 e 1 B) = (B 1 e,A} = - (e,B 1 A) .// With this Theorem and Lemma, the restricted Moufang identities follow. THEOREM 6.3. Let e be the adjoined basis element used to form B. For any A,B in B, the following restricted Moufang identities hold: i} (eAe)B = e[A(eB)] ii) B(eAe) = [ (Be)A]e iii} (eB} (Ae) = e (BA) e. Proof. i) (eAe)B-e[A(eB)] = (eA 1 e,B)+(e,A,eB) = -(e,eA,B)-(e,eB,A) = -[e(eA)]B+e[ (eA)B]+e[ (eB)A]-[e(eB}]A = -(e2A}B-(e2B)A+e[(eA)B+(eB)A] = AB+BA+e[ (eA)B+(eB)A] = e[-e(AB)-e(BA)+(eA)B+(eB)A] = e[ (e,A,B)+(e,B,A)] = 0. The proof of ii) is similar. [ (Be)A]e-B(eAe) = (Be,A,e)+(B,e,Ae} = -(B,Ae,e)-(A,Be,e) = -[B(Ae)]e + 32 B[(Ae)e]-[A(Be)]e+A[(Be)e] = B[Ae2 ]+A[Be2 ]-[B(Ae)+A(Be)]e = -BA-AB-[B(Ae)+A(Be)]e = [ (BA)e+(AB)e-B(Ae)-A(Be)]e = [(B,A,e)+(A,B,e)]e = 0. For iii), consider: (eB) (Ae)-e (BA)e = (e,B,Ae) + e[B(Ae)-(BA)e] = -(e,Ae,B) - e(B,A,e) = -(eAe)B + e[(Ae)B-(B,A,e)]=-e[A(eB)] + e[(Ae)B-(B,A,e)] = -e[A(eB)-(Ae)B+(B,A,e)] = -e[-(A,e,B)+(B,A,e)] = -e[(B,e,A)+(B,A,e)] = 0.// COROLLARY 6.4. (B,eA,e) = -(B,e,A)e. Proof. (B,eA,e) = [B(eA}]e- B(eAe) = [B(eA)]e- [(Be)A]e = [B(eA)-(Be}A]e = -(B,e,A)e.// Considering the Cayley-Dickson multiplication, a question is why (a1+ea2 ) (b1+eb2 ) = (a1b 1-b2i 2 > + e THEOREM 6.5. (cf, [28], p.46). Let A be an alterna- tive algebra with 1, and with involution, and let the adjoined element e be such that e 2 = -1 and ae = ea for all a in A. Then the polynomial multiplication in A + eA is the Cayley-Dickson multiplication. Proof. Following Schafer's proof, first note that the alternative laws imply that, for all a,b in A, the follow- ing hold: 33 i) a(eb) = e(ab) ii) (ea)b = e (ba) iii) (ea) (eb) = -ba. The remainder of the proof is easy.// Hence, up to the octonions, the Cayley-Dickson multi plication and the polynomial multiplication are the same. Actually, the restriction that A be an alternative algebra is stronger than necessary. THEOREM 6.6. Let A be any Cayley-Dickson algebra. Let e be the adjoined basis element used to construct B = A + eA. Assuming that we have already defined in B a dis tributive multiplication, let e satisfy: i) e 2 = -1 ii) ae = ea- £or all a in A iii) (e,a,b) + (a,e,b) = 0 for all a,b in A. Then, for all a,b in A, we have: i) a(eb) = e(ab) ii) {ea)b = e(ba) iii) (ea) (eb) = -ba. Proof. We first notice that assuming (e,a,b) + (a,e,b) = 0 for all a,b in A gives us Lemma 6.2, Theorem 6.3, and Corollary 6.4 for a,b in A. Now to see i), con sider: o = - (e,a,b) - (a,e,b) = {e,a,b) + (a,e,b) = (ea)b- e(ab) + (ae)b- a(eb) = (ea)b- e(ab) + (ea)b a(eb) = Ie(T(a)-a)]b + (ea)b- a(eb) - e(ab) = T(a) (eb) - (ea)b + (ea)b- a(eb) - e(ab) = IT(a)-a] (eb) - e(ab) = a ( eb ) - e ( ab ) • 34 Similarly, for ii), 0 = (a,b,e) + (a,e,b) = (a,b,e) + (a,e,b) = (ab)e- a(be) + (ae)b- a(eb) = e(ba) + (ae)b a(be-eb) = e(ba) + (ae)b- a(be-be) = e(ba) + (ae)b T(b)(ae) =e(ba) + (ae)(b-T(b)) =e(ba) + (ae)b=e(ba) + (ea)b. To show iii), we need one of the restricted Moufang identities. (ea) (eb) = (ea) (be) = e (ab)e = e[ (ab)e] = COROLLARY 6.7. Under the same hypotheses as given in Theorem 6.6, the Cayley-Dickson multiplication in B is the same as polynomial multiplication in B. Proof. a 1b 1 + (ea2 ) (eb2 ) + a 1 (eb2 ) + (ea2 )b1 = a 1b 1 + (-b2a2) + e(alb2) + e(bla2).// It must not be supposed, however, that ae = ea is true for arbitrary 8lements in the algebra B. In particu- lar we have: LEMMA 6.8. Let x = a 1+ea2 • Then xe =ex if and only if T(a2 ) = 0. -- Proof. (a1+ea2 )e = a 1e+ea2e = -a2+ea1 whereas e(al-ea2) = eal-e(ea2) These are equal if and only if a 2 = -a2 .;; This leads one to ask under what conditions the con- elusions of Theorem 6.6 are true for arbitrary a and b in B, assuming the Cayley-Dickson multiplication in B. LEMMA 6.9. Let A,B be elements of B, where A= a 1 +ea2 and B = b 1+eb2 • Then, i) A(eB) = e(AB) if and ~nly if T(a2 ) = 0 35 ii) (eA)B = e(BA) if and only if T(AB) = T(b2 ) = 0 iii) (eA) (eB) = -BA if and only if T(AB) = T(b2 ) = 0. Proof. Since Theorem 6.1, Lemma 6.2, and Theorem 6.3 are true for A,B in B, we need only note that in the proof of Theorem 6.6 for i) we must have Ae = eA to get A(eB) = e(AB) which by Lemma 6.8 means T(a2 ) = 0. Similarly, to show (eA)B = e(BA), we examine the proof of ii) in Theorem 6.6, and see we need (AB)e = e(BA) and eB =Be. Thus again the result follows by Lemma 6.8. For (eA) (eB) = -BA, the proof of iii) in Theorem 6.6 requires eB = Be and (AB)e = e(BA).// We now wish to examine the basis elements themselves. As stated before, every element in the Cayley-Dickson An can be written in the form r.r.e., i = 0, ••. ,2n-1, algebra 1 1 1 where r. is real, e = 1, and e. 2 = -1 otherwise. The 1 0 1 basis {e. I i = 0, ••• ,2n-l} arises from the basis of A 1 by 1 n- the old basis and adjoining one new element e retaining 2n-1 and all multiples of it by the old basis elements. So, if {e0 , ••• ,e _1 } is the basis for A 1 and the adjoined 2n -1 n- basis element is e n-l' then the basis for An is {e0 ,e1 , ••• , 2 } 0 < J. < 2n-l_l. e 1 ,e 1 , ••• ,e.e 1 , ••• ,e wereh 2n- -1 2n- J 2n- 2n-l The Cayley-Dickson multiplication demands that we call e.e 1 = e 1 • We are now able to observe several J 2n- 2n- +j interesting properties that are possessed by these basis elements. In the following, e. is a basis element of A • 1 n 36 LEMMA 6.10. Basis elements obey: i) e.e. = e .e., if i or j = 0, or if i = j ~ J J ~ -- ii) e.e. = -e .e., if i,j I 0 and i I j • ~ J J ~ Proof. If i or j = 0, or if i = j, the result is obvious. Otherwise e. and e. may be each of two types, ~ J ( ek + e• 0) or (0 -eek ) f, or 0 < k < 2n-l wh ere e = e • 2n-1 This gives rise to four cases. Since the result is clearly true for quaternions, it will then follow by indue- tion. Case 1: i,j < 2n-l; follows from the induction hypo- thesis. n-1 . 2n-l Case 2: i < 2 , J ~ ; here e. = (e.+e·O) and ~ ~ e. = (0-eek), for some k < 2n-l. Now e.e. = (e.+O) (0-eek) J ~ J ~ = -e(eiek) = e(eiek), and ejei = (0-eek) (ei+O) = -e(eiek). Case 3: i > 2n-l, j < 2n-l; this proof is similar to Case 2. Case 4 : ~,J· · ~ 2n-l ; h ere ei = (0 -eek ) d an ej = (0 -eem ) n-1 for some k,m < 2 • Then eiej = (0-eek) (0-eem) = -emek = emek, and ejei = (0-eem) (0-eek) =-~em=- ekem = emek.// Although not all Cayley-Dickson algebras are alterna- tive, their basis elements do satisfy the alternative prop- erty, as is shown by the following theorem proved by Schafer [26]. THEOREM 6.11. e. (e.e.) =-e.= (e.e.)e. fori I 0. ~ ~ J J J 1 1 --- Proof~ The proof is essentially the same as Schafer gives, and is given here for the reader's convenience. By 37 suffices to show e. (e.e.) =-e .• The result is true for the l. l. J J octonions, since they are alternative. Thus the proof is by induction on the dimension of A • Assume the statement n previous Lemma, there are four true for An- 1 • As in the cases. n-1 Case 1: i,j < 2 ; the result holds by the induction hypothesis. n-1 Case 2: i < 2 1 j ~ 2n-l; here e. = (e.+O) and 1. 1. n-1 e. = (0-eek), for some k < 2 • Now e. (e. e.) = J l. 1. J (e.+O) [ (e.+O) (0-eek)] = (e.+O) [O+e(e.ek)J = e[e. (e.ek)] = l. 1. 1. l. 1. 1. -e[ei (eiek)] = -e(-ek) = eek = -ej. n-1 n-1 Case 3: i ~ 2 , j < 2 ; here ei = (0-eek) and n-1 e. = (e.+O) for some ek < 2 • Then e. (e. e.) = J J 1. 1. J (0-eek) [ (0-eek) (ej+O)] = (0-eek) (O-e[ejek]) = - (ejek)ek = 2 (ejek)ek = ejek = -ej. . . 2n-l Case 4: l.,J ~ i here ei = (0-eek) and ej = (0-eem) n-1 for some k,m < 2 • Then e. (e. e.) = (0-eek) [ (0-eek) (0-ee ) ] 1. l. J m = ( 0-ee ) ( -e e +0) e[ (emek)ek] = -e[em(ek2 )] eem k m k = = = -e .. I I J COROLLARY 6.12. If ei is any basis element of An' and if xis rlnv element of A, then (e.,e.,x) = 0. ~ -- n ---- 1. l. n Proof. Let x = I.r.e., j = 0, .... , 2 -1, r . real. By J J J J the linearity of the associator, (e.,e.,x) = I.r.(e.,e.,e.} 1. 1. J J 1. 1. J = o. I I Notice, however, that Corollary 6.12 does not imply (e.,e.,x) = -(e.,e.,x), since this would imply that 1. J J 1. e. (e.x) =-e. (e.x), whenever e.e. = -e.e .• This is false l. J J 1. 1. J J 1. 38 in general, as the following example shows. In any Cayley Dickson algebra containing A4, we have (e1-e15,e:1-e15 ,e4} = (el,el,e4} - (el,el5'e4) - (el5'el,e4) + (el5'el5'e4) = - THEOREM 6.13. if i,j ~ 0 --and i = j I or if i ~ 0, j = o. e.e.e. = -- l. J l. e.' if i = o, or if i,j F 0 J - - - and i F j. Proof. The result is immediate if i = j and i,j F 0, or if i F 0 and j = 0. For the other situation, the proof is by induction on the dimension of An. n-1 Case 1: i,j < 2 ; this follows by the induction hypothesis. Case 2: i < 2n-l, j ;:: 2n-l; here ei = (ei+O) and ( } f k < 2n-l. Thus e. e . e. = I e. ( 0-eek)] e. = ej 0-eek , or l. J l. l. l. [ -e ek) l e. = -e[ei (eiek)] = e[ei (eiek)] = -eek = ej. (e l.. l. . n-l . n-1 Case 3: l. .:: 2 , J < 2 ; here ei = (0-eek) and n-1 e . = (e. +0) , for k < 2 • So e.e.e. = I (0-eek)e.] (0-eek) = J J l. J l. J -[e(ejek)] (0-eek) = -ek{ejek) = -ek(ekej) = ej. Case 4: i, j > 2n-l; here ei = {0-eek} and ej = (0-eem), n-1 for k ,m < 2 • So e. e .. e. = I (0-eek} (0-eem) J (0-eek) = l. J l. (-emek) (0-eek) = e[ (emek}ekl = ei (ekem)ek] = -e[ (ekem}ek] = -eem = ej, since if k = m, then ei = ej' a contradiction.// 39 We are now able to prove a theorem which may seem obvious, but the proof of which involves several cases. THEOREM 6.14. {e . e . ) ek = +e . {e . ek ) • ]. J - ]. J Proof. The result is clear if i,j,k is 0, or if any two basis elements are the same. Therefore, assume that each is not the identity and no two are the same. The proof is by induction on the dimension of An and there are eight cases. In the proof, great use is made of Theorem 6.6, and the observation that the induction hypothesis and Lemma 6.10 allow {eiej)ek to be rearranged and reassociated in any order, with only perhaps a sign change, as long as i,j,k are less than 2n-l. n-1 Case 1: i,j,k < 2 i this follows by the induction hypothesis. n-1 n-1 Case 2: j,k < 2 , i ~ 2 ; here ei = erne, where rn < 2n-l. Then {e.e.)ek = [ (e e)e.]ek = +[ (ee )e.]ek = 1 J m J - m J +[e{e.e )]ek = +e[ek(e.e )]. Likewise, e. (e.ek) = - J m - J m 1 J {e e) ( e . ek) = + ( ee ) ( e . ek) = +e [ ( e . ek) e ] • m J - m J - J m Case 3: i,k < 2n-l, j > 2n-l; here the proof is similar to Case 2. case 4: i,j < 2n-l, k > 2n-l; this case is similar to Case 2. Case 5: i J. > 2n-l k < 2n-l,. here e. = e e and e. = ' - ' 1 rn J e e, for rn,p < 2n-l. Therefore (e.e.)ek = [ (e e) (e e)]ek = p 1 J m p ± n-1 n-1 Case 6: i,k > 2 , j < 2 ; this case is similar to Case 5. n-1 n-1 Case 7: j,k > 2 , i < 2 ; this case is similar to Case 5. h C ase 8 : 1,],· · k ~ 2n-l ; ere e. = e e, e. = e e, and 1 m J p n-1 e = e e, for some m,p,q < 2 Then ( e . e . ) ek = k q 1 J [ (e e) (e e)] (e e) = +(e e } (e e) = +(e e )e , and e. (e. ek) = m P q - p m q - p m q 1 J +(e e) (e e } = +e (e e ).// (erne) [ (epe} (eqe)] = - m qp -m qp LEMMA 6.15. For basis elements ---of the octonions: e. (e. ek}, if i,j, or k = 0, 1 J or if i = j , j = k, or i = k, (e. e.) ek = -- 1 J or if e.e. = +_ek. -- 1 J -e. (e.ek)' otherwise. 1 J Proof•. The proof for (e. e.) ek = e. (e. ek) under the 1 J 1 J hypotheses is easy. It should be noted, however, that e.e. = +ek is equivalent, because of the alternative 1 J - property, to the statement that the product of any two is plus or minus the third. The proof of the last situation involves carefully reviewing the proof of the previous theorem with e = e 4 ./! For one application of this material, we consider the following definition. Definition 6.1. The anticomrnutator ]x,y[ is defined to be ]x,y[ = xy + yx. let A= \.a.e. and B = I.b.e. for For the following, l1 1 1 1 1 1 where a. and b. are real. i=O, ••• ,n, 1 1 41 LEMMA 6.16. AB + BA = 2[a0b 0 - Iiaibi]e0 + 2L.((a b.+a.b )e.], i = l, ••• ,n. l l l l 0 0 Proof. Recall that e.e. = -e.e. if i,j f 0 and if j, l J J l and e.e. = e.e. otherwise. Consider the multiplication l J J l chart for AB given below. b e boeo blel b2e2 n n +a b e aoeo aoboeo +aOblel +a0b2e2 0 n n 2 alel albOel +alblel +alb2ele2 +albnelen 2 a2e2 a2b0e2 +a2ble2el +a2b2e2 +a2bne2en a e +a b e 2 n n n n n A multiplication chart for BA would be similar, but with the entries in reverse order. Thus, for each summand a.b.e.e. in AB, there is a summand b.a.e.e. in BA. Hence, lJlJ JlJl all terms for AB + BA cancel, except those of the first row, the first column, and the diagonal. From the AB table, we obtain I . (a b - a.b.)e , I.a b.e., and I.a.b e. from 1 0 0 l l 0 l l l 0 l l l 0 the diagonal, the first row, and the first column. We obtain similar sums from the BA table, except the results of the first row and the first column are interchanged. Thus, adding all those terms which do not cancel, the result of the lemma is obtained.// THEOREM 6.17. ]A,B[ = 0 if and only if T(A) = T(B) = 0 and \.a.b. = 0, i = l, ••. ,n, where A,B f 0. -- L1 1 l Proof. From Lemma 6.16, AB + BA = 0 implies 42 If a 0 = 0, then aibO = 0 for all i implies ai = 0 for all i or else b = 0. But a. = 0 for all i contradicts A ~ 0. 0 1. Thus T(B) = 0. On the other hand, if 0, then b. = ao ~ 1. [ -aibO ]/ao so that a 0b 0 + I i a i [ (a i b 0 ) I a 0 l = o. Then 2 a 2) bo Now by Lemma 6.16, if T(A) = T(B) = 0, then AB + BA = -2[I.a.b.]e • Hence AB + BA = 0 implies T(A) = T(B) = 0 1. 1. ~ 0 and L. a . b . = 0. 1. 1. 1. Conversely, if T(A) = T(B) 0 and r.a.b. = 0, then = 1. 1. 1. Lemma 6.16 implies AB + BA = 0.// 43 VII. Zero Divisors. In this section, we consider zero divisors in the Cayley-Dickson algebras. First we consider norms, and then use the properties of norms to examine zero divisors. A. A. Albert [2] defines an absolute-valued algebra to be an algebra over the reals with a function f: A + R, such that f(O) = O, f(a) > 0 if a~ 0, and f(ab} = f(a}f(b). He proves that the octonions are the only absolute-valued nonassociative algebra. He also defines a normed algebra in a similar way, but requires only f(ab}~ f(a}f(b). Albert then shows that every real algebra is a normed algebra under f =I· !r. 1, where a= I.r.e., with e. a l l l l l l basis element of the algebra, i = l, ••• ,n. In any Cayley-Dickson algebra of dimension higher than eight, for the norm N(a} = aa, we have N(a} = 0 if and only if a= 0, N(a) > 0 for a~ 0; but, in general, N(ab) ~ N(a)N(b), since if a and bare mutual zero divisors, N(a)N(b) ~ 0 and N(ab) = 0. What is striking, though, is that under this norm, these algebras are not even normed algebras. Consider the following example from A4 : Let A = e 1+e10 and B = e 0+e1 +e4-e15• Then AB = -e0+e1+e10+e11• Hence, N(A) = 2, N(B} = 4; but N(AB) = 4 • i . e. , N (A) N (B) > N (AB) • Now, let A = e 1-e10 and B be as before. Here, AB = -e0+e1+2e5-e10-e11+2e14 • Thus, N(A) = 2, N(B) = 4; but N(AB) = 12. i.e., N(A)N(B) < N(AB). 44 The question then arises as to what exactly is the relationship between N(AB) and N(A)N(B). THEOREM 7.1. In any Cayley-Dickson algebra N(A)N(B) - N(AB) = [N(a1 )N(b1 )-N(a1b 1 )] + [N(a1 )N(b2 )-N(a1 b 2 )] + [N(a2 )N(b1 )-N(a2b 1 )] + [N (a2 ) N (b2 ) -N (a2b 2 ) 1 + T [ (a1 b 1 ) (a2E2 )- (a1 b 2 ) (a2E1 )], where A = a 1+ea2 and B = b 1+eb2 • Proof. Recall that N(A) = N(a1 ) + N(a2 ). Therefore N(A)N(B) = [N(a1 )+N(a2 )] [N(b1 )+N(b2 )] = N(a1 )N(b1 ) + N(a2 )N(b1 ) + N(a1 )N(b2 ) + N(a2 )N(b2 ). on the other hand, AB = (a1b1-b2a 2 > + e(a1b 2+b1a 2 ), so N(AB) = N(a1b 1 ) + N(b2a 2 )- T[(albl) (b2a 2 )] + N(a1b 2 ) + N(b1a 2 > + T[(a1b 2 > (b1a 2 >l = N(a1b 1 > + N(a1b 2 > + N(b2a 2 > + N(b1a 2 > + T[(a1b 1 ) (a2E2 >- COROLLARY 7.2. In A4 , N(A)N(B) - N(AB) = T £ ca1 b1 >ca 2E2 >- ca 1b 2 > ca 2E1 > 1 • Proof. Here, a.,b. are octonions and hence, N(a.b.) = l. J l. J N (a. ) N (b. ) • I I l. J Note that now an alternate proof of N2 (A) = N(A2 ) in 2 - - A4 is obtained by observing that T[N(a2 >a1 -(a1a 2 > (a2a 1 )l = 0, because of the Moufang identities in A3 • A major difference between A4 and the octonions is the existence of zero divisors in A4 • We now wish to examine zero divisors in detail. In the following, assume that A 45 and B are mutual zero divisors, A and B are elements of A , n and that A = a 1+ea2 , B = b 1+eb2 • THEOREM 7.3. The following are equivalent: i) AB = 0 ii) BA = 0 iii} AB = 0 iv} A-lB = 0 v) (eA} (eB) = 0 vi} (eA)B = 0 vii) A(eB) = 0. In addition, the above imply: viii} T(A} = 0 ix) T (a1 } = 0 x) T (a2 } = 0 xi) A and B are linearly independent. Proof. i) +-+ ii} This follows from the facts that N(A) = 0 if and only if A= 0, and N(AB) = N(BA). i) + xi) Recall that there are no nilpotent elements in A Let rnA + nB = 0, for m,n real. Then, mA2 + nAB = 0 n implies mA2 = 0. This implies m = o. Similarly for n. i} +-+ iii) In any flexible ring with involution N(AB) = N (AB) . Thus the following are all equivalent: AB = 0, N(AB) = 0, N(AB) = 0, and AB = 0. iii} +-+ iv) This follows from the identity A-lB = [AB] /N (A). - i) + viii) AB = [T(A)-A]B- AB, so AB = 0 implies AB = 0 and T(A)B = 0. Thus T(A) = 0. 46 viii) + ix) This follows from the fact that for A = a 1 +ea2 , T(A) = T(a1 ). iii) ++ v) By the restricted Moufang identities, (eA) (eB) = 0 if and only if 0 = (eA) (eB) = (eA) (Be) = e(AB)e if and only if AB = 0. v) + x) eA = -a2+ea1 • Hence T(eA) = T(a2 ) = 0. ii) ++vi) Since T(AB) = T(a ) T(a ) T(AB) = 1 2 = = T(b1 ) = T(b2 ) = 0, the result follows from Lemma 6.9, i.e., {eA)B = e(BA). iii) ++ vii) By the same reasoning as above, Lemma 6.9 gives A(eB) = e(AB).// LEMMA 7.4. The following are equivalent: i) T (A) = 0 ii) A- = -A iii) A-l = -A/N(A) iv) A 2 = -N (A). Proof. l, ... ,n. Moreover, T(A) = 0 means r 0 = 0. Hence, i), ii), and iii) are equivalent. To see iv), recall that A2 - T(A)A + N(A)l = 0.// We now find it desirable to define an isomorphism, * from A to A in the following way. n n Definition 7.1. ~A* = a 1 -ea2 if A = a 1 +ea2 . The * isomorphism gives some added information about zero divisors, but first we need to consider some properties of the * operator. 47 THEOREM 7.5. The following properties hold for the * operator. i) A + A* = 2a1 ii) A* = A ---if and only if a 1 is real. iii) A* = A if and only if a 2 = o. iv) T (A*) = T (A) v ) N (A* ) = N (A) vi) (A+B) * = A*+B* vii) (AB) * = A*B* viii) (rA)* = rA*, r real ix) (eA)* = -eA* x) [A,B]* = [A*,B*] xi) (A,B,C)* = (A*,B*,C*) xii) (A)* = A* xiii) A** = A. Proof. i) A +A* = (a1 +ea2 )+(a1-ea2 ) = 2a1 • ii) A* = al-ea2 = al-ea2 =A if and only if al = al' i.e. , if and only if al is real. iii) A* = a 1-ea2 = a 1 +ea2 = A if and only if a2 = o. iv) T(A*) = A* + A* = (a1 -ea2 )+(a1+ea2 ) = al+al = T (A). v) N(A*) = A*A* = (a1-ea2 ) (a1+ea2 ) = a 1a 1+a2a 2 = N(A). vi) (A+B)* = [ (a1+b1 )+e(a2+b2 )]* = (a1+b1 )-e(a2+b2 ) = (a1-ea2 )+(b1-eb2 ) = A*+B*. vii) (AB)* = [ (a1b 1-b2a 2 )+e(a1b 2+b1a 2 )]* = (a1b 1-b2a 2 ) -e(a1b 2+b1a 2 ) = (a1-ea2 ) (b1-eb2 ) = A*B*. viii) (rA)* = r*A* = rA*. ix) (eA)* = e*A* = -eA*. 48 x) [A,B]* = (AB-BA)*= A*B*-B*A* = [A*,B*]. xi) {A,B,C)* =I (AB)C-A(BC)]* = (A*B*)C*-A*(B*C*) = (A* , B* , C * ) • xii) (A)* -ea > * -a +ea {a --ea ) A*. = {a 1 2 = 1 2 = 1 2 = xiii) A** = (a -ea ) * a +ea = 1 2 = 1 2 A.// We are now able to consider * operators and zero divisors. LEMMA 7.6. AB = 0 if and only if A*B* = 0. Proof. Since from the definition of A*, it is clear that A* = 0 if and only if A = 0, we have 0 = AB = (AB)* = A*B*.// Definition 7.2. Let (A,B) be the vector subspace generated by elements A and B. Now, we consider the structure of the subspace THEOREM 7.7. If A is a zero divisor, then [-N(A)]n/2 , if n is an~ integer. ([-N(A)] (n-l)/2 )A, if n is an odd integer. Proof. First we observe that the result is true for n an even or odd positive integer. This follows from the 2 fact that A = -N(A). [-N(A)]n/2 • If n is odd, An= (An-l)A = ([-N(A)] {n-l)/2 )A. Now if n is zero, the result is obvious. If n is negative, the An= (A-l)-n = [-A/N(A)]-n = {[-1/N(A)]-n)A-n. Hence, if n is even, An= 49 [-1/N{A)]-n[-N(A)]-n/2 = I-N(A)](-n/2 )+n. If n is odd, An= [-1/N(A)]-n[-N(A)] (-n-1)/2 = (I-N(A)] [ (-n-l)/2J+n)A.// Note that, since AB = 0 if and only if BA = 0, the results of Theorem 7.7 hold true forB as well. THEOREM 7.8. If A and Bare mutual zero divisors, then: [-N(A)]n/2 [-N(B)]m/2 , n,m even integers. 0 , n,m odd integers. ([-N(A)]n/2 [-N(B)] (m-l)/2 )B, n even, m odd. ([-N(A)] (n-l)/2 [-N(B)]m/2 )A, n odd, m even. Proof. Simply multiply An by Bm using the results of Theorem 7.7.// THEOREM 7.9. Let A and B be mutual zero divisors, then: r +r A+r B, r. real, i) Proof. i) This is clear by Theorem 7.8. ii) This follows since the basis elements A and B of A satisfy the are commutative, and since all elements n Jordan identity. iii) This follows since, A2B = -N(A)B ~ A(AB) = 0.// Thus, the vector subspace generated by any two mutual zero divisors is a three dimensional, non-alternative, commutative Jordan algebra with zero divisors. Now, let us go back to the nature of the zero divisors themselves. Our objective is to determine just when an so element A in A will be a zero divisor. n Lill~ 7.10. AB = 0 is equivalent to any pairing of an equation i) or ii) with an equation iii) or iv), where: i) a 1b 1+b2a 2 = 0 iii) b1a 2-a1b 2 = 0 ii) b 1a 1+a2b 2 = 0 iv) a 2b1-b2a 1 = 0. Proof. Recall T(a) = 0 implies a= -a. Then AB = 0 if and only if equations i) and iii) hold; B(eA) = 0 if and only if equations i) and iv) hold; BA = 0 if and only if equations ii) and iii) hold; and (eA) B = 0 if and only if equations ii) and iv) hold.// Definition 7.3. The antiassociator )A, B,C ( is defined by )A,B,C( = (AB)C + A(BC). LEMMA 7.11. The antiassociator is linear in each argument. Proof. The proof is similar to that of showing the associator is linear in each argument.// We may now prove the major theorem of this section. THEOREM 7.12. Let A= a 1+ea2 and B = b1+eb2 , where A,B ~ 0. Let a 1 ~ 0 and a 2 ~ 0 have no zero divisors. Then AB = 0 if and only if i) and ii) hold, where: i) )a1 ,b1 ,a2 ( = -(a1 ,a1 ,b2 )+(a2 ,a2 ,b2 )+[N(a2 )-N(a1 )lb2 ii) (a1 ,b1 ,a2 ) = (a1 ,a1 ,b2 )+(a2 ,a2 ,b2 )+[N(a2 )+N(a1 ) ]b2 . Proof. If AB = 0, then (a1b1+b2a 2 ) = 0 and (a1b 2-b1a 2 ) = 0 by Lemma 7.10. Thus a 1b 1 = -b2a 2 and a 1b 2 = b 1a 2 • Thus (a1b 1 )a2 = -(b2a 2 )a2 and a 1 (a1b 2 ) = a 1 (b1a 2 ). Adding gives (a1b 1 )a2 + a 1 2 2 -(al,al,b2)+(a2,a2,b2) + (a1 -a2 )b2 , using the flexible property. Now, recall T(a) = 0 implies N(a) = -a2 • Thus, >a1 ,b1 ,a2 ( = -(a1 ,a1 ,b2 )+(a2 ,a2 ,b2 )+IN(a2 )-N(a1 )]b2 • Similarly, (a1b 1 )a2-a1 (b1a 2 ) = -(b2a 2 )a2 - a 1 (a1b 2 ) = 2 2 -(b2 ,a2 ,a2 )-b1a 2 +(a1 ,a1 ,b2 )-a1 b 2 • Thus, (a1 ,b1 ,a2 ) = (a1 ,a1 ,b2 )+(a2 ,a2 ,b2 )+[N(a2 )+N(a1 )]b2 • Conversely, it is easy to see that equation i) implies (a1b 1 )a2+a1 (b1a 2 ) = a 1 (a1b 2 )-(b2a 2 )a2 and that equation ii) implies (a1b 1 )a2-a1 (b1a 2 ) = -(b2a 2 )a2-a1 (a1b 2 ). Now, if we add these two equations, and then subtract them, we get 2(a1b 1 )a2 = -2(b2a 2 )a2 and 2a1 (b1a 2 ) = 2a1 (a1b 2 ). i.e., (a1b 1+b2a 2 )a2 = 0 and a 1 (b1a 2-a1b 2 ) = 0. Now if a 1 and a 2 have no zero divisors, then a 1b 1+b2a 2 = 0 and a 1b 2-b1a 2 = 0. Hence, by Lemma 7.10, AB = 0.// THEOREM 7.13. Let A and B be mutual zero divisors. Then: i) T([a1a 2 lb1 ) = T([a1a 2 ]b2 ) = 0 ii) a 1 = ±a2 implies a 1b 1 = a 1b 2 = 0 iii) a 1 and a 2 are not nonzero real numbers iv) a 1 = 0 implies a 2b 1 = a 2b 2 = 0 v) a 2 = 0 implies a 1b 1 = a 1b 2 = 0. Proof. i) From a 1b 1 = -b2a 2 and b 1a 1 = -a2b 2 , we see that (a1b 1 )a1 = -(b2a 2 )a1 and -a1 (b1a 1 ) = a 1 (a2b 2 ). Adding, and using the flexible property, we obtain 0 = a 1 (a2b 2 )- (b2a2)al = al (a2b2)- al (a2b2) = al (a2b2)+al(a262) = manner, we also T(a1 [a2b 2 ]) = T([a1a 2 ]b2 ). In a similar have (alb2)al = (bla2)al and al(a2bl) = al(b2al). 52 Subtracting and using the flexible property gives 0 = a 1 (a2b 1 )-(b1a 2 )a1 , and the proof follows as before. ii) I af 1 = a 2 , t h en b a 1 1+b2a 1 = 0 and a 1b 1-b2a 1 = 0. Thus, since T(a1 ) = T(b2 ) = 0, and b 2a 1 = a 1b 2 , we obtain a 1b 1 = a 1b 2 = 0. If a 1 = -a2 , then a 1b 1-b2a 1 = 0 and a 1b 1+b2a 1 = 0; and the result follows as before. iii) This follows directly from T(a1 ) = T(a2 ) = 0. iv) and v) follow directly from AB = 0 if and only if a 1b 1+b2a 2 = 0 and a 1b 2-b1a 2 = 0 if and only if b 1a 1+a2b 2 = 0 and b 2a 1-a2b 1 = 0.// We now consider how zero divisors behave in A4 • THEOREM 7.14. Let A be ~zero divisor in A4 . Then: i) a 1 ~ ~a 2 ii) a 1 and a 2 are not real numbers iii) N(a1 ) = N(a2 ). Proof. i) This follows from Theorem 7.13, ii), and the fact that a.,b. are octonions and have no zero divisors. l l ii) This follows from Theorem 7.13, iii) and the fact that if a. were zero, Theorem 7.13, iv) and v), implies A l or B = 0. iii) Recall that the octonions are an absolute-valued algebra, i.e., N(ab) = N(a)N(b). Also, in any Cayley Dickson algebra, N(a) = N(-a) = N(a) ~ 0. Thus, AB = 0 if and only if a 1b 1+b2a 2 = 0 and a 1b 2-b1a 2 = 0. Hence N(a1 )N(b1 ) = N(b2 )N(a2 ) and N(a1 )N(b2 ) = N(b1 )N(a2 ). Dividing, we obtain [N(a1 )]/N(a2 ) = [N(a2 )l/N(a1 ). Thus, 2 2 N (a1 ) = N (a2 ) and the result follows.// 53 We are now able to prove our most useful result for finding zero divisors in A4 • THEOREM 7 .15. A and B are mutual ze·ro divisors· in A4 if and only if the following three conditions hold: i) N (a1 ) = N (a2 ) ii) b 2 = [(a1b 1 )a2 ]/N(a1 ) iii) )a1 ,b1 ,a2 ( = 0, where A = a 1+ea2 and B = b 1+eb2 • Proof. If AB = 0, then by Theorem 7.12 and the fact that octonions are alternative, we obtain )a1 ,b1 ,a2 ( = [N(a2 )-N(a1 )Jb2 and (a1 ,b1 ,a2 ) = [N(a2 )+N(a1 )Jb2 • Now by Theorem 7.14, iii) N(a2 ) = N(a1 ). Thus, )a1 ,b1 ,a2 ( = 0 and (a1 ,b1 ,a2 ) = 2N(a1 )b2 • Now 2(a1b 1 )a2 = (a1 ,b1 ,a2 ) + )a1 ,b1 ,a2 ( = 2N(a1 )b2 • Thus, the result follows. Conversely, if N(a1 ) = N(a2 ), then b 2 = [(a1b 1 )a2 1/N(a1 ) implies [N(a2 )+N(a1 )Jb2 = 2(a1b 1 )a2 = (albl)a2-al(bla2)+(albl)a2+al(bla2) = (al,bl,a2) + )a1 ,b1 ,a2 ( = {a1 ,b1 ,a2 ) by iii). Finally, N(a1 ) = N(a2 ) and )a1 ,b1 ,a2 ( = 0 imply )a1 ,b1 ,a2 ( = [N(a2 )-N(a1 )Jb2 . Now since by the alternative property, {a1 ,a1 ,b2 ) = (a2 ,a2 ,b2 ) = O, the result follows by Theorem 7.12.// COROLLARY 7.16. If A and Bare mutual ~divisors, then the following holds in A4 : i) No three of a 1 ,a2 ,b1 , and b 2 ~be quaternions. ii) Each of ~ following antiassociators is zero: )al,bl,a2(; )bl,al,b2(; )al,b2,a2(; )bl,a2,b2(; )a2,bl,al (; )b2,al,bl (; )a2,b2,al {; )b2,a2,bl {. 54 Proof. We first prove ii) by noting that the follow- ing are all equivalent: AB = 0, BA = 0, (eA)B = 0, B(eA) = 0, (eA) (eB) = 0, (eB) (eA) = 0, (eB)A = 0, and A(eB) = 0. Thus the order of the ai in a 1+ea2 and the bi in b 1+eb2 may be permuted. i) If any three were quaternions, they would be associative, contradicting part ii).// Considering the above, one would hope to be able to look at a particular A f 0, at least in A4 , and determine whether or not it is a divisor of zero by inspection. In fact, we do know that if A = a 1+ea2 is in A4 , and if N(a1 ) f N(a2 ), a 1 = ~a 2 , T(a1 ) or T(a2 ) t 0, or either a 1 or a 2 real, then A is not a zero divisor by Theorem 7.14, and Theorem 7.3. Each of the necessary and sufficient conditions for zero divisors, however, involved considering B as well as A. The following example illustrates that putting all the restrictions on A that we have encountered as necessary (without considering B) still doesn't guaran- tee that A will be a divisor of zero. Yet letting b I.r.e., 1 = J. J. J. i = 0, •.. ,7, and computing )a1 ,b1 ,a2 ( = 0 yields b 1 = 0. Hence B = 0 and A is not a zero divisor. Thus, it appears that there is no way to look at a specific A and be sure if it is a zero divisor, unless you actually solve for B such that AB = 0. There are two obvious ways to do this. The first would be to set 55 B I.b.e., i 0, ••• ,2n-1. Then after calculating AB = = 1 1 1 = I.c.e., i 0, ••• ,2n-1, one could set each c. = 0 and solve. 1 1 1 = 1 Unfortunately, the multiplication could be extremely long, and solving c. = 0 could result in solving 2n-l equations 1 in 2n-l unknowns. The author used a PL-1 computer program to aid in working several examples in this way. This program, which multiplies arbitrary Cayley-Dickson elements in A4 , is included in Appendix C. Even a computer is little help in solving the system of equations when its solutions are not rational, however. The other way suggested by our previous work is to n-1 set b I.r.e., i 0, ••• ,2 -1, and solve equations i) 1 = 1 1 1 = and ii) of Theorem 7.12. In A4, this reduces the problem to, at worst, seven equations in seven unknowns. The aid of a computer in performing the multiplication is useful, even here. Solving the equations remains tedious, however. It seems apparent that what is needed is further investiga- tion into the antiassociator. From here, we concentrate on in , where the elements a.,b. acted on by zero divisors A4 1 1 the antiassociator will be octonions. LEMMA 7.17. If a,b,c are octonions, then )a,b,c( = 0 implies T(a) = T(b) = T(c) = 0. Proof. By Theorem 7.15 (ab)c+a(bc) = 0 implies (a//N(a} + e 8 [c/IN{c)]) is a zero divisor. Hence, T(a/IN(a)) = 0. Thus T(a) = 0. The other parts are similar.// 56 We now consider for which basis elements of the octonions the antiassociator vanishes. THEOREM: 7.18. For basis elements of the octonions, )e.,e.,ek( = 0 if and only if 1. J i) None of i,j, or k is 0, ii) iii) e.e. =I ±ek. 1. J Otherwise, )e.,e.,ek( = 2(e.e.)ek. 1. J 1. J Proof. Recall Lemma 6.15.// THEOREM 7.19. For basis elements of the octonions, if i,j,k =I 0, then: i) )e.,e.,e.( = )e.,e.,e. ( = -2e., 1. 1. J J 1. 1. J ii) { -2ej, if i = j I )e.,e.,e.( = - 1. J 1. 2e., if i =I j I J - iii) ) e. , e., ek ( = - ) ek, e . , e . {. 1. J J 1. Proof. i) Since basis elements are alternative, we 2 2 have )e. ,e. ,e. ( = e. e. + e. (e.e.) = 2e. e. = -2e .• 1. 1. J 1. J 1. 1. J 1. J J ii) Recall Theorem 6.13. iii) If i,j,k =I 0, then -)ek,e.,e. ( = -(eke.)e.-ek(e.e.) J 1. J 1. J 1. =-e. (e.ek)-(e.e.)ek =e. (e.ek)+(e.e.)ek = )e.,e.,ek{.// 1. J 1. J 1. J 1. J 1. J Since we are interested in antiassociators of elements each of which has trace 0, the condition that i,j,k =I 0 in Theorem 7.19 is no restriction to us. ~HEO~EM 7.20. Let e.,e.,ek be basis elements of any 1. J - Cayley-Dickson algebra. Then )e.,I.r.e.,ek( = 0 implies 1. J J J )e.,e.,ek( = 0 for all j = o, ... ,n. 1. J ------57 Proof. Notice first, that since (e.e.)ek =+e. (e.ek) J. J - J. J by Theorem 6.14, )e1 ,ej,ek( =either 0 or 2(eiej)ek. Now by linearity we may write )e. ,I.r.e.,ek( as I.r.)e.,e.,ek(. l J J J J J l J Suppose not all these terms are zero, say r )e.,e ,ek( 1 0, m 1 m m = l, •.. ,h. Then 0 = I r )e. ,e ,ek( m m 1 m = Imrm·2(eiem)ek = 2[e.(I r e )]ek. 1 m m m But Corollary 6.12 says (A, ek, ek) = 0, hence if Aek = 0, then 0 = (Aek) ek = A(ek2) = +A. Likewise, if e.B = 0, B = 0. Thus I r e = 0 which is a contradiction l mmm of the linear independence of the em's.// With the apparatus above, we often may calculate all the zero divisors of a given number in A4 quickly since many, if not most, antiassociators will vanish. In fact, by Theorem 7.18, since we assume i,j,k 1 0, )e. ,e.,ek( = 0 l J unless i = j, j = k, or i = k, or e.e. = ±ek. The only l J time we even need to consult the basis multiplication table is for that last case, eiej = ±ek. Consider the following example. Let A= e 1+e8e 2 • We B such that AB 0. Let b = I.r.e., i = 0, .•. ,7. seek = 1 l l l Since T(b1 ) = 0, r 0 = 0. Since T([a1a 2 Jb1 ) = 0, T([e1e 2 Jb1 ) = T(e3b 1 ) = 0, implying r 3 = 0. Now expanding 0 = )e ,I.r.e. ,e ( and discarding any antiassociator which 1 l l l 2 is zero, we obtain 0 = r 1 )e1 ,e1 ,e2 C+r 2 >e 1 ,e2 ,e2 ( = -2r1e 2-2r2e 1 • Thus r 1 = r 2 = 0. Hence b 1 = r 4e 4+r5e 5+ r 6e 6+r7e 7 • Now letting b 2 = [ (a1b 1 )a2/N(a1 ) = [e1 (r4e 4+r5e 5+r6e 6+r7e 7 )Je2 we get b 2 = r 7e 4-r6e 5+r5e 6-r4e 7 • 58 Hence, B = (r4e 4tr5e 5tr6e 6tr7e 7 )te8 (r7e 4-r6e 5tr5e 6-r4e 7 ) = r 4 The last thing we do in this section is examine the equation AX = B and its solutions. LE~~ 7.21. In~ alternative algebra with involution, -1 AX = B, A ':/- 0, always has A B ~ a solution. Proof. Since A(A-lB) = -(A,A-l,B)+B = -(A,A/N(A},B) + -1 B = (A,A,B)/N(A) + B = B, we conclude that A B is always a solution.// This seemingly obvious choice of a solution need not work in A4 , however. THEOREM 7.22. In A4 , we have the following: -1 A(A B)= B + [(a1 ,b2 ,a2 )-e(a1 ,b1 ,a2 )]/N{A}, where A= a 1+ea2 and B = b 1+eb2 • Proof. A{A-lB) = B + {A,A,B}/N{A), and by Theorem 4.3, {A,A,B) = (a1 ,b2 ,a2 }-e(a1 ,b1 ,a2 ) for A,B in A4.;; It is clear that only certain equations will have the solution A-1B. However, even then, the solution may not be unique. For example, if A has a zero divisor C, so that AC = 0, then A{X+C} = B if AX = B. The situation can be worse, however, because for some equations of this type, there are no solutions. Consider the following example. Let A= es+elS and X= Iiriei, i = 0, ••• ,15. Then a little 59 calculation shows AX = (-r5-r15 )e0 + (r5-r15 >e10 - (r4+rl4) (el+ell) + (r7+rl3)e2 + (r7-rl3)e8 + (rl2-r6) (e3+e9) + (rl-rll) (e4+el4) + (rO-rlO)es + (rO+rlO)elS + (r3+rg) (e6-el2) - (r2+r8)e7 + (r8-r2)el3" Notice the pairing of e 1 and e 11, e 3 and e 9 , e 4 and e 14 , and e 6 and e 12 • It is clear that no choice of ri will let AX be any single element of any of these pairs. In par ticular, (e5+e15 )x = e 1 has no solution in A4 • An unproven conjecture is that if AX = B has a solu tion, then X = A-lB is a solution. In particular, all zero divisors for (e5+e15 ) must be of the form: m(e1+e11 ) + n(e3-e9 ) + p(e6+e12 ) + q(e4-e14 ), m,n,p,q real. Hence, another unproven conjecture is that if A has no zero divisor, then AX = B has the unique solution X = A-1B. 60 VIII. Examples and Counter Examples. In the first part of this section, we wish to find the set of zero divisors of a given nonzero A in A4 • Then we wish to find all zero divisors of that set. In general, this is extremely complicated; but if A is of the form A= e.+e8e., 1 < ~, j _< 7, the situation results in seven ~ J - disjoint systems of interrelated zero divisors. In the following, let {A,B} be the set of all linear combinations of A and B. Also, let {X} ++ {Y} indicate that xy = 0 for all x in X and for all y in Y. THEOREM 8.1. Each A of the form ei+e8ej which has ~ zero divisor is in one and only one system ~elow, and all of its zero divisors ~ in the same system. System 1: System 2: System 3: 61 System 4: System 5: {el+el4'e6-e9}~(------~){e2+el3'e5-elO}~{e3+el2'e4-ell} . {el-el4'e6+eg}< >{e2-el3'e5+el0} {e3-el2'e4+ell} System 6: System 7: {e2+ell' e3 -elO H ){ e4 ~el3' es+el2 }><{ e6+el5' e7-el4} {e2-ell'e3+el0}~(~---;>{e4+el3'e5-el2} {e6-el5'e7+el4} Proof. To see that this theorem is true, one must compute all zero divisors of elements of the form e.+e e .• 1 8 J Although this is quite lengthy, it is easy to do using the methods of the last chapter.// Recall that A* = a1-ea2 if A = a1+ea2 • Then notice that each system above has the following form: 62 Thus we may find all zero divisors of the element e 1+e10 , for example, by looking at system 1, and noting that all zero divisors of m1 (e1+e10 ) + m 2 (e2-e9 ) are of the form nl (e4-el5) + n2(e7+el2) + n3(e5+el4) + n4(e6-el3), where n. and m. are arbitrary real numbers. ~ ~ Consideration of the examples above suggests the following. THEOREM 8.2. Let A be a fixed nonzero element in A4 • Let B = {BI AB = O} and e 8 B = {e8Bi B is in 8}. Then B is an additive group closed under multiplication £y e 8 • Proof. First e 8 B is contained in B since (e8B)A = 0 if and only if AB = 0. Similarly, B is contained in e 8 B since B = -e8 (e8B). Thus B is closed under multiplication by e 8 • To see that B is an additive group, consider that AB1 = 0 and AB 2 = 0 implies A(B1+B 2 ) = 0 and the fact that AB = 0 implies A(-B) = 0.// It is not true, in general, that B.B. is in B when B. ~ J l and B. are in B. For example: Let A = = J el+elO' Bl e4-el4' and B2 = es+el4" Then ABl = AB 2 = 0. But B1B2 = 2(el-el0) so that A(B1 B2 ) = 4e11 ~ 0. An unanswered question remains as to whether or not there exists A,B,C in A4 such that AB = AC = BC = 0. From the systems of zero divisors above, it is clear that such A,B,C do not exist of the form e.+e e.. If such A,B,C do ~ 8 J exist in any Cayley-Dickson algebra, then the subspace generated by A, B, and C would be a subalgebra. 63 Another unanswered question also deals with mutual zero divisors. Let B = {BI AB = 0}. Then we have observed that (e8A)B = 0. Now does XB = 0 imply X= 0, X= A, or X= e 8A? If A= e.+e e., the answer is yes, in view of the systems 1 8 J above. In fact, for every example the author has considered, the answer is in the affirmative. Earlier, we observed that AB = 0 implies that (A,B) is a commutative Jordan algebra with zero divisors. We now consider an example of the nature of (A,B) where B = {BI AB = 0}. Let A = e 4-e15. Then considering system 1, we see that B = {r1 B1+r2B2+r3B3+r 4B4 }, where B1 = e 1+e10 , B2 = e 2-e9 , B3 = e 5-e14 , and B4 = e 6+e13 , where ri is an arbitrary real number. Note that B is a four dimensional subset of A4 • In every example of dozens worked out, the same thing occured. For A= e.+e e., the systems above 1 8 J show that B is always four dimensional. Now consider the set generated by A and B. It is easy 2 to verify that -A /2 = e 0 , (A-[B1 B3 ]/2)/2 = e 4 , (A+[BlB3]/2)/2 = elS' BlB2/2 = e8, e4e8 = el2' el5e4 = ell' e 8e 15 = e 7 , and e 4e 7 = e 3 • Moreover, the following table shows closure under multiplication for these basis elements. 64 eo eo -e -e e3 e3 0 4 -e e4 e4 0 -e -e e7 e7 0 8 -e e8 e8 -elS 0 -e -e -e ell ell el2 3 0 7 -e -e el2 el2 -ell 4 3 -e -e -e elS elS e8 7 4 0 Thus, (A,B) is an eight dimensional subalgebra of A4• Note that (A,B) is clearly not isomorphic to the octonions since it contains zero divisors. The following result is now apparent. LEMMA 8.3. Not all eight dimensional subalgebras of A4 ~ isomorphic to the octonions. One of the original questions which prompted this paper, was how AS differed from A4. Some answers to this can now be considered. First of all, recall from Theorem 7.5, i), that if A= a 1+ea2 is in An' then A+ A* is in An-l· THEOREM 8.4. (A+A*,A+A*,B+B*) = 0 for all A,B in A4, but (A+A*,A+A*,B+B*) 1 o for all A,B in As. Proof. A3 is alternative, A4 is not.// As was seen in the last chapter, a major difference between A4 and A5 is in the nature of their zero divisors. Let us recall here these differences. If A and B are 65 mutual zero divisors, then the following hold for A,B in A4, but do not necessarily hold for A,B in A5 : i) a 1 ~ ±a2 , ii) a 1 and a 2 must be nonzero, iii} N(a1 ) =N(a2). The fact that each of these holds for zero divisors in A4 was shown in Theorem 7.14. The fact that they do not hold in A5 is easy to see. For, let a be an element of A4 0 (as in such that for b1 and b2 in A4 we have ab1 = ab 2 = the example immediately following Theorem 8. 2) • Now, letting A = (a+e16a) and B = (bl +el6b2) ' we see that A and Bare elements of A5; but AB = 0, contradicting i). Simi larly let A = (O+e16a) and B be as before. Then AB = 0, contradicting ii) and iii). Finally, it was mentioned in the beginning of this paper that many results have been obtained in non associative algebras by considering "defining identities". All algebras satisfying a particular identity, or group of identities, are classed together and then studied. Examples of this which we have seen are the flexible algebras, the alternative algebras, and the noncornmutative Jordan algebras. Several such classes of algebras are well known, and others appear only recently in the literature. Each is an attempt to restrict a broad class of algebras such as the noncommutative Jordan algebras to a smaller, more fr.ui tful, class of algebras. In order to proceed, we need the following definitions. 66 Definition 6.1. A finite-dimensional algebra is said to be a standard algebra if it satisfies the identities: i) (x,y,z) + (z,x,y) - (x,z,y) == 0, ii) (w,x,yz) - (wy,x,z) - (wz,x,y) = 0, for all w,x,y,z in the algebra. This definition was first given by A. A. Albert [3]. Definition 6.2. A nonassociative algebra over a field F of characteristic ~ 2 is a generalized standard algebra A if: i) A is flexible, ii) H(x,y,z)x = H(x,y,xz), where H(x,y,z) = (x,y,z) + (y 1 Z 1 X) + ( Z 1 X 1 y) 1 iii) (x,y,wz) + (w,y,xz) + (z,y,xw) = [x, (w,z,y)] + (x,w,[y,z]), 2 iv) if F has characteristic 3, then (x,y,x ) = 0, v) y [x (wz)] -x [y (wz)] + (x, wz, y) + [ (wz) x] y- [ (wz) y] x = [y(xw)-x{yw)+(x,w,y)+(wx)y-(wy)x]z+ w[y(wz)-x(yz)+{x,z,y)+(zx)y- (zy)x], for all w,x,y,z in A. Schafer first gave this definition in 1968 [29]. Definition 6.3. An accessible algebra is defined by the identities: i) (x,y,z) + (z,x,y) - (x,z,y) = 0, ii) (Iw,x] ,y,z) = 0. These were first given by E. Kleinfeld [18]. Definition 6.4. A generalized accessible algebra satisfies the identities: 67 i) [x, (z,y,y)] = 0, ii) 3(x,y, [w,z]) = -[w, (x,y,z)]-2[x, (y,z,w)J+ 2[y, (z,w,x)]+[z, (w,x,y)]. These were first defined by E. Kleinfeld, M. H. Kleinfeld, and J. F. Kosier [69]. The best account of the interrelation between these ideas is given in the last mentioned paper, and is condensed in the following diagram, where---+ stands for implication: ~ JORDAN ASSOCIATIVE~ COMMUTATIVE > STANDARD~ ALTERNATIVE ACCESSIBLE GEN. STANDARD~ ~GEN. ACCESSIBLE/ We now wish to show, by means of counter examples, that not only the Cayley-Dickson algebras are not generalized accessible; but also that they do not satisfy any of the defining identities given above, except (x,y,x) = 0 and (x,y,x)2 = 0, i.e., the flexible property and the noncommu- tative Jordan identity, respectively. Proceeding through the identities in the order given, the standard identities are not satisfied. i) If X = es, y = e8' z = elO' then (x,y,z} + (z,x,y) - (x,z,y) = 6e7 ~ 0. ii) If X = el, y = el+elO' z = e4-el5' w = e7+el2' so that yz = yw = o, then (w,x,yz) - (wy ,x, z) - (wz,x,y) = 0 - 0 - (2e8,e1 ,e1+e10 > = -4e3 ~ 0. The generalized standard identities ii), iii), and v) do not hold. 68 ii) Let x = e 1 , y = e 15 , z = e 4 • Then H(x,y,z)x = -2e11 , but H(x,y,xz) = 2e11• iii) Let x = y = e 1 , w = e 15 , z = e 4 . Then all terms in (x,y,wz)+(w,y,xz)+(z,y,xw) = [x,(w,z,y)] + (x,w,[y,z]) are zero except (x,w,[y,z]) which is 4e11• v) Let x = e 5-e14 , y = e 5+e14 , z = e 4-e15 , w = Then y[x(wz)]-x[y(wz)]+(x,wz,y)+[(wz)x]y-[(wz)y]x = 0. On the other hand: [y(xw)-x(yw)+(x,w,y)+(wx)y-(wy)x]z +w[y(xz)-x(yz)+(x,z,y)+(zx)y-(zy)x] = -2e4 - 4e5 + 4e14 + 2e15 'f 0. The accessible identities do not hold. i) (x,y,z) + (z,x,y) - (x,z,y) = 0 was one of the standard identities. ii} ([w,x] ,y,z) = 0 doesn't hold, for let w = e 1 , x = e 2 , y = e 5 , and z = e 8 . Then ([w,x] ,y,z) = (wx,y,z) (xw,y,z) = (e3 ,e5 ,e8 )-(-e3 ,e5 ,e8 ) = 2(e3 ,e5 ,e8 ) = -4e14 'I 0. The generalized accessible identities fail to hold. i} Let x = e 3 , y = e 1-e15 , z = e 4 • Then [x, (z,y,y)] = [e 3 ,-2e10 J = -4e9 'f 0. ii) Let x = z andy= w. Then 3(x,y, [w,z]) = -[w, (x,y,z)]-2[x, (y,z,w)]+2[y, (z,w,x)]+[x, (w,x,y)] becomes 3(x,y, [y,z]) = 0 because of the flexible property. Letting xy = -yx yields (x,y,[y,x]) = -2(xy) 2+2x[y(xy)], so letting x = e 1-e15 andy= e 7+e13 , we obtain (x,y,[y,x]) = 4(e4+e10 ) :f 0. Richard Block 17], in 1969, further generalized the generalized standard algebras by considering noncommutative 69 Jordan algebras that also satisfy what he calls alterna tivity conditions. He gives the following definitions. Definition 6.5. An algebra A is completely alternative if (x, [y,z] ,w) + (x,w, [y,z]) = 0, for all x,y,z,w in A. Definition 6.6. An algebra A is semicomEletely alter native if ([x,y],z,z) = 0, for all x,y,z in A. Definition 6.7. An algebra A is strongly alternative if ([v,w], [x,y],z) + ([v,w],z, [x,y]) = 0, for all x,y,z,v,w 1.n A. We now show that the Cayley-Dickson algebras do not satisfy any of these identities. The Cayley-Dickson algebras are not completely alter native. If w = x andy= x, then (x, [y,z],z) + (x,x, [x,z]) = (x,x, [x,z]) by the flexible property. Now, let z = x = e 1-e15 , y = e 4 • Then (x,x,[x,z]) = 4(e5-e11 ) ~ 0. The semicompletely alternative identity fails by letting z = x = e 1-e15 , andy= e 4 . For ([x,y],z,z) = -4(e5-e11 ) ~ 0. Finally, the strongly alternative identity fails. For, if v = e 9;2, w = e 13 , x = e 1/2, y = e 14 , and z = e 1 , we have ([v,w],[x,y],z)+([v,w],z,[x,y]) = (e 4 ,e15 ,e1 ) + (e4,el,el5) = 2el0 ~ 0. Thus, not only are the Cayley-Dickson algebras not members of any of the algebras defined above, not one of the defining identities holds in the sixteen dimensional Cayley-Dickson algebra. 70 IX. Conclusions. In this paper we have examined the structure of zero divisors in real Cayley-Dickson algebras. A necessary and sufficient condition for zero divisors in a real Cayley Dickson algebra of any dimension was found. Antiassociators played a major role in the description of zero divisors. It was found that the use of an IBM 360 computer was very valuable in finding examples and in checking hypotheses. It appears that computers hold much promise in the further study of higher dimensional Cayley-Dickson algebras. Finally, we have observed that Cayley-Dickson algebras are a class of noncommutative Jordan algebras which are integer power associative. The fact that A4 satisfies none of the defining identities of the preceding chapter is significant. It means that there remains much work to be done in classifying noncommutative Jordan algebras. It appears no one has considered as a class noncommutative Jordan algebras which are integer power associative. The author feels that favorable results could be obtained from such a study. 71 BIBLIOGRAPHY [ 1] A. A. Albert, Quadratic forms permitting composition, Ann • of Hath. 4 3 ( 19 4 2 ) , 161-1 7 7 • [2] ,.--...,....,...---=-,_....,..,.....,...' Absolute-valued real algebras, Ann. of Math • 4 8 ( 19 4 7 ) , 4 9 5-5 0 1. -- [ 3] ~~-~--~' Power-associative rings, Trans. Amer. Math • Soc • 6 4 ( 19 4 8 ) , 5 5 2-59 3 • [ 4] =-----~~-~' A theory of trace-admissible algebras, Proc. Nat. Acad. sci. U.S.A. 35 (1949) I 317-322. [5] =-~--_,_ _ _,__, Absolute-valued algebraic algebras, Bull. Amer. Math. Soc. 55 (1949), 763-768. [ 6] ~--~----' On Hermitian operators over the Cayley algebra, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 639- 640. [7] Richard E. Block, Noncomrnutative Jordan algebras with commutators satisfying ~ alternativ1ty condition-,-- Proc. Nat. Acad. Sci. U.S.A. 65 (1970), 10-13. [ 8] R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87-89-.- --- [ 9] H. s. M. Coxeter, Integral Cayley numbers, Duke Math. J o 13 (1946) 1 561-578. [10] Charles W. Curtis, The four and eight square problem and division algebras, MAA studies in mathematics, Volume 2 (ed. A. A. Albert), Prentice-Hall, Englewood Cliffs, N.J., 1963, 100-125. [11] L. E. Dickson, On quaternions and their generaliza tions and the history of the eight square theorem, Ann. of Math. 20 (1919), 155-171. [12] s. Eilenberg and E. Niven, The fundamental theorem of algebra for quaternions, Bull. Amer. Math. Soc. 50 (1944) 1 246-248. [13 J D. Finkelstein, Josef Jauch, S. Schiminovich, and David Speiser, Foundations of quaternion quantum mechanics, J. Math. Phy. 3 (1962), 207-220. [14] H. H. Goldstine and L. P. Horwitz, On a Hilbert space with nonassociative scalars, Proc. Nat. Acad. Sci. u.s.A. 48 (1962), 1134-1141. 72 [15] ~-----.---:--r------=~--=-~~--' Hilbert s7ace with nonassociative scalars I, Math. Annalen 154 1964), 1-27. [16] , Hilbert sp(ce with -n~o,....n~a~s-:-s-o-:-c-~'1"":·a~·-=-t"""~-:-v-e-· -s~c-a~l~a-:-r-s-:-.-.~I~I-,__,..M,....a....,t..,.h-.-Anna1en 16 4 19 6~ 291-316. [17] J. Jamison, Extension of some theorems of complex functional analys~s to-rinear spaces over the quatern~ons, Ph.D. DISsertation, university of Missouri at Rolla, 1970. [18] Erwin Kleinfeld, Standard and accessible rings, Canad. J. ~-lath. 8 (1956), 335-340-.- [19] Erwin Kleinfeld, Margaret Humm Kleinfeld, and J. Frank Kosier, The structure of generalized accessible rings, Bull. Amer7 Math. Soc.-r5 {1969), 415-417. [20] A. G. Kurash, General algebra, Chelsea Publishing Co., New York, 1963. [21] Kevin McCrimmon, Norms and noncornmutative Jordan algebras, Pacific J. Math. 15 {1965), 925-956. [ 22] ------,~~---=--' Structure and representation of ~ commutat~ve Jordan algebras, Trans. Amer. Math. Soc. 121 (1966}' 187-199. [2 3] John Milnor, Some consequences of ~ theorem of Batt, Ann. of Math. 68 {1958), 444-44~ [ 24] A. J. Penico, On the structure of standard algebras, (Abstract) Pro~ Internat. Congress of Math., Cambridge, Mass. 1 (1950), 320: Amer. Math. Soc., Providence, R.I., 1952. {25] R. Penny, Octonions and the Dirac equation, Arner. J. of Physics 36 (1968), 871-873. [ 26] R. D. Schafer, On the algebras formed~ the Cayley Dickson process, Amer. J. of Hath. 76 {19ffi, 435-446. [ 2 7] , Noncommutative Jordan algebras of -c~h-a_r_a_c~t-e_r_i~s-t,....lr.c Q, Proc. Amer. Math. Soc. 6 {1955), 472-475. I2 81 , An introduction to nonassociative _a_l_g_e~b-r_a_s_,__ A_c_a-dernic Press, New York and London, 1966. [ 2 9] , On ~eneralized standard algebras, Proc. ~N-a-t-.~A-c-a~d-.~8-ci. U.S.A. 60 (1968), 73, 74. 73 [30] , Generalized standard algebras, J. of Algebra 12 (196 9) , 386-417. [ 31] , Forms pe·rm:i tting composition 1 Advances in Math. 4 ( 19 7 0 ) 1 12 7-14 8 • [32] Maria J. Wonenburger, Automorphisrns of cayley algebras, J. of Algebra 12 (1969) 1 441-452. 74 VITA Harmon Caril Brown, the son of Reuben R. and Lela M. Brown, was born in Grand Blanc, Michigan on June 19, 1941. After attending elementary and secondary school at Grand Blanc, Michigan, he attended Harding College, Searcy, Arkansas where he received the B.S. degree in mathematics in June, 1963. While at Harding, he was active in inter collegiate debating and was president of the student body. There he married the former Sarah Ellen Colvin of Vivian, Louisiana. They now have three children: Mark, Sarah, and David. After a summer at Louisiana Polytech, Ruston, Louisiana, he taught 7th and 8th grade mathematics in Bernie, Missouri the school year of 1963-1964. Then he attended Michigan State University, receiving the M.A. degree in mathematics in March, 1966. From September 1966 to 1972, he was instructor of mathematics at the University of Missouri at Rolla, where he received the Ph.D. degree in mathematics in May, 1972. 75 Appendix A BASIS MULTIPLICATION TABLE FOR OCTONIONS eo e1 e2 e3 e4 e5 e6 e7 eo eo e1 e2 e3 e4 e5 e6 e7 -e -e -e -e e1 e1 0 e3 2 e5 4 7 e6 -e -e -e -e e2 e2 3 0 e1 e6 e7 4 5 -e -e -e e3 e3 e2 1 0 e7 6 e5 -e4 -e -e -e -e e4 e4 5 6 7 0 e1 e2 e3 -e -e -e -e e5 e5 e4 7 e6 1 0 3 e2 -e -e -e -e e6 e6 e7 e4 5 2 e3 0 1 -e -e -e -e e7 e7 6 e5 e4 3 2 e1 0 76 Appendix B BASIS MULTIPLICATION TABLE FOR A4 eo el e2 e3 e4 e5 e6 e7 e8 e9 elO eo eo el e2 e3 e4 e5 e6 e7 e8 e9 elO -e -e -e -e -e el el 0 e3 2 e5 4 7 e6 e9 8 -ell -e -e -e -e -e e2 e2 3 0 el e6 e7 4 5 elO ell 8 -e -e -e -e e3 e3 e2 1 0 e7 6 e5 4 ell -elO e9 -e -e -e -e e4 e4 5 6 7 0 el e2 e3 el2 el3 el4 -e -e -e -e e5 e5 e4 7 e6 1 0 3 e2 el3 -el2 el5 -e -e -e -e e6 e6 e7 e4 5 2 e3 0 1 el4 -el5 -el2 -e -e -e -e e7 e7 6 e5 e4 3 2 el 0 el5 el4 -el3 -e -e e8 e8 9 -elO -ell -e12 -el3 -el4 -el5 0 el e2 -e -e -e e9 e9 e8 -ell elO -e13 el2 el5 -el4 1 0 3 -e -e -e elO elO ell e8 9 -el4 -el5 el2 el3 2 e3 0 -e -e ell ell -elO e9 e8 -el5 el4 -el3 el2 3 2 el -e -e el2 el2 el3 el4 el5 e8 9 -elO -ell 4 es e6 -e -e el3 el3 -el2 el5 -el4 e9 e8 ell -elO 5 4 e7 -e -e -e el4 el4 -el5 -el2 el3 elO -ell e8 e9 6 7 4 -e -e -e el5 el5 el4 -el3 -el2 ell elO 9 e8 7 e6 5 77 Appendix B continued BASIS MULTIPLICATION TABLE FOR A4 ell el2 el3 el4 elS eo ell el2 6 13 el4 el5 el elO -el3 el2 el5 -el4 -e e2 9 -el4 -elS el2 el3 -e e3 8 -elS el4 -el3 el2 -e -e e4 el5 8 9 -elO -ell -e es -el4 e9 8 ell -elO -e e6 el3 elO -ell 8 e9 -e -e e7 -el2 ell elO 9 8 e8 e3 e4 es e6 e7 -e -e e9 e2 5 e4 e7 6 -e -e -e 6 10 1 6 7 e4 es -e -e -e 6 11 0 7 e6 5 e4 -e -e -e -e 6 12 e7 0 1 2 3 -e -e -e 6 13 6 el 0 e3 2 -e -e el 6 14 es e2 3 0 -e -e -e 6 15 4 e3 e2 1 0 78 Appendix C PL-1 PROGRAM FOR CHARACTER STRING MULTIPLICATION CAYLEY-DICKSON ALGEBRA ELEMENTS SKC: PROCEDURE OPTl(;f\:) (Mfl.l."') • DC l . I'll ' ~f l (*).CHAR ( lOC) 1/.:~K. CuNl Fdll tl) , lll>.1 f!XEO, H F l XED_, 512(*) CHAt{(l00) V1-\F< C'!NlPULLED ~ FtXEO , ' ZPCJ (HI... f:: ( 2) Sl](*) CH~~~db4) VA~<:.. '.. J•:~-:DfJLU:.fJ, CfllQ U~~.k.( 15) VAP. , UJl FIXFf1, LU(O:t5,v: l5) FIXED, XYZ LHARC100) Vf~Y!NG , PRJD(*,*I, t..HI\k(l.;'J) Vt." tCNTR.UILFD, NU~(*) J:'Jf~-~U~~ ·~>:,_,59 1 l>~,JRuLLEO, ~·!U2(*1 PlLJUI-E ·~_,S~'J' C:.•~:-,R;_,LLED, F fiXED N' U 3 C* ) P i C l U P t: ' ~ S ::; ~ '1 1 r U r..; l h: (j L L E0 , NUK(*) PICTUt-.E •::;:l:::0~9' ,uitvfROLLfD, (M,O,I,K,JJL'M~,~Y,iX,~,!~t~O,lK,IQ) FIXE[) , DU CHAR(80 Vt.!', TN C HA '' ( l ) CAT OiL· 1\ { 6 4 J V r R , S CHAR(l), NUEl(*l FIXED CONf~ULL~D , NUF2C*) FiXEn ((~fkOLLfO , NUE3(*1 FTX[f1 CCI\ITkCLLlll, ~UM(!l*~ PlC.TU~E •ss~~Y' l~~7~LLLED , ~tR P1C.U~E ')~J$9' , S A V ( H \ F ( 6 4 ) V t1 P , N E E (* ,* ) F l X FD CJ N .f t-. ~ , L L E D , CZ,Y,X,W 1 V,UJT'~~'R'. PpA.) FIXED , C ( * ) CHAR ( J 2 V /'.: '..., C •, l'< i t • :::,L L!:. 0 , TFMP CHAR ( 100) Vt\P.. , OUT CHARC2J0) VAR , fEII.1F FIXFD , T[~L Fi.XED ; I* All ELFMENT SHGULD Bl PUT lN /. l~rw DATA f~kO AND EJ'.Lrl PAi-,1 SEPARAlfl) ~y CuMfviAS IN FkONT 1JF EACH DATA ~ECl!CIJ ;~DATA (kti-D ~HOULD BE ru:.CE::D WH { C H C 0 NT A 1 N ') THE NU ~ 8 E "< r' F P td~ 1 t, N 0 1 ~ f N T H F. N U ~ H E F<, U F ELEMFNT IN EACH PA~l·. lHESt NU~B(qS SHUULO BE RTGH1 JU~TIFlED iN A 5 COLUMN fiFLO. ALL PFMA~NING 5 COLUMN FJ.ELD!:> SHOULD AE FILL CY ·:. BEFORE PUNNING THIS PRQGP.t..M C''·JF r-'U~T F1ND CUT THE LARGFSl LFNGTH !1F ANY OUTPUT lLP~t:Nl ,;,'W Tt!Er>.. CHANGE "!'HE FiELD PRCDC* *) t..HAR(---) VA~ CCNTROLLED LUC~TED IN THE O~CLA~~ SIAr~MET~ TO FIT THIS ~L[M[Nl I lS THE NUMAfR OF POLY~C~lAL K IS lHF NUMRE::P GF ELEME~I~ ,~ THE Fl R.~ T PCL YI.DMI Al J lS THf NUMRER UF ELEMt:~lS !N THt SE.Cr'r~~ POLYNOMl AL l I S 1 Hf NU ~BE h OF ELF M E f\ 1 ~ I t\ l H l THIF.U PlJLYhi(IMIAL Y I S r H F t\J U M 8 E R 0 F E L E ,., F 1\. -, ~ 1 ~~ ., H t FCURTH POLYNOMIAL ~ IS THf ~UMBE~ OF ELE~ENT~ t~ lHf FIFTH PIAYNuMIAL W I S T Ht N UM R E !=: C' F E: L Et-1 H n S i :--.: f H~ S l X·r H P r:: L YN :_, M I A L 79 V IS THE f'..IUMBF.R OF ELEt-',Et-..TS ih flH: SEVFNH1 PL~LYNOMIAL U iS THf NUMBFR OF ELEMENTS IN THE EIGHTH POLYNDMl~l T IS THE NUMBEP CF Elt:MENTS IN lHf Nl"JTH PULY~LMIAL SS I~ THE NUMBE~ GF ELEMfNTS IN lHt TENTH POLYNCMIAL ~ lS THE NU~PER DF ELEMEt-..15 iN IHE ELEVEN1H P~LVNG~lAL H I~ THI: !~UMbER uF ELI::MEI\1:> 1N THF TWELFTH POLYNf'MlAL B T$ lHE NUMBEI~ C:F lLFMfNTS ,~.T\1 "IHI-: THJRlE.FNTH POlY''4CMlto~l ~ 15 lHE NUMBFP. OF ELLI-lFN"f!:J lN "IHt FOIJRlHTEENlH PULYNClMI~- SAV=SURSTR(STllMZ),U!I-2,21; TN=SUBS1PlSAV,2,l) ; 00 Q = 0 BY 1 TU q ; PUT STRING($) EDIT (01 (F(1.~ll IF fN=S THEN Dfl ; NUE1(Ml)=IJ ; GO TO JQ ; END ; ENO ; JQ: TN=SU~STRCSAVfl,l) ; DO Q = 0 BY 1 0 q ; PUT STklNG(S) EDIT (Q) (F(1.0)) IF TN=S THEN on ; NUEl(fol:l)=NUE11•"1l)+O*lO; GO TO J QQ ; FND ; END ; JQ(~: Ir: 00-5-0=0 THEN :>Tl(MZ)=' t ; EL:, E S T 1 ( ~z )= ~U R!:. l R ( S T l ( 1-'J l , C H, U'J- 5-;J) END ; i<.f: ZOZ=ZQZ+l ; PUl STRINGIZPQ) ~Oil lZQll (F(2,~)) CHZQ=•POLYNCtMIAL I II ZPQ ; PUT PAGE LIST CCHZ:)); OG MZ = 1 BY 1 TC J ; GFT EOlT (ST2(~Zl,OUl (A{64),A(l6)) 0=0 ; NlJ2CMZl=O ; PL'f SKIP DATA (Sf2(MZll ; IF ~UBSfPlST2l-.,Zl,l,U='l' SUASTR(ST2(MZ),l,l)='2' S'JBSTPlST2P~Z), 1,11='-' SUBSTR(ST2(MZ),l,l)='3' SUBSTR(ST2(MZ),l,ll='4' SUB~TR(Sl2(~l),l,l)='5' SUBSTRCST2l~Z),l,ll='6' SUBS1R(ST2(MZ), 1,1)= '7' SURSTR($12(MZ),l,ll= 1 8 1 SUBSTRl~T2(MZ),l,ll='9' S~BSTRlST2(MZl,l,ll= 1 0 1 THEN OG ; DO CJ = 1 R Y 1 l L fA· WH 1 l [( .) UR :;) 1 F ( S I 2 ( :'1Z ) , L , l) -. =1 , ' ) ENO ; tAT=SU~SlR(ST2(MZl,l,J-l) ; IH=-1 ; 00 IX= LFNGlH(CtT) BY -1 TO l ; IH=IH+l ; IF IX=l & ~UBSHI.fC.A-IrlrU='-' IHE:I-.. DC: NU7(MZ)=-1*NU2(MZ) ; GL TO QQ(,) ; ENO ; TN=SUBSTR(CAT,IX,1) ; DO Q = 0 BY l TO q ; PUT STRING(S) FOil (Q) (F(l.D)) IF TN=S THEN OG ; NU2(~l)=NU2(M7)+0*10**JH GO TO JUt~P2 ; END ; END ; JUMP2: END ; Q~~Q: END ; ab 5 dGN~ 2 tM§~=l fo 64 WHiLUSUBSTtdST2(MZl,CU,l)-.=' 'l END ; ~AV=SUBSTk(~T2tMZ),00-2,2) ; TN=SUBSTR(~AV,2,1) ; DO Q = 0 BY 1 TO 9 ; PUT STklNG(S) EDIT (Q} (F(l.U)) IF fN=S THEN DG ; NUE2(MZ}=Q ; GL.l TO JQ2; END ; 81 END ; JQ2: TN=SUBSTR(SAV,l,l) DO Q = 0 AY 1 TO q ; PUT STRING(S) EDIT (Q) (F(l.U)) IF TN=S THEN DO ; NUf2(MZ)=NUE2(MZ)+n*l0 GO TO JQQ7 ; END ; END ; JOQ7: IF 00-5-U=O THE"J Sl2{MZ)='' ; E l S E S l 2 ( Ml ) =S t J 8 S T k ( S T 2 ( Ml l , (l + 1 , ;J L;- 5- ') ) END ; I* MULTIPLY *I DO MZ = 1 BY 1 TU K ; 00 MY = 1 BY 1 TO J ; IF L=l THFN SAB=LU(NUE2(MY),NUEl(Ml)) f L SE SA B = L U ( NU E 1 ( Ml ) , NU E 2 (MY) ) ; IF SAB FREE NEE . ALLOCATF. PROD(K,J)' ALLOCATE NUM(K,J) ; ALLOCATE NEE(K,J); GO TG R E ; END ; FREE ST 1 • FREE S l2 i FREE NUl ; FREE NU2 ; FPEE NUEl ; FREE NUE2 ; M= K*J ; ALLOCATE STlCMJ ALUJCA TE NUl (M) ; ALLOCATE NUEU~l; IK=O; gg ~~ ~ l ~~ l t8 ~ ~ IK=IK+l ; ST1CIK)=PROOCMZ,MY) ; ~H~1~~~;~~~~~~!~~~,;; END ; END ; MM=M*32 ; PUT SKIP DATA01fMMJ; ~LtOCATE C ~MJ ; rnJ ~Z=l BY 1 TO M ; IQ:l ; IK=l ; DO 0 = 1 BY l TO LENGTH(Sl"l tMZ)) ; IF SUBSlRCSllPH),O,l)=•,• Tt.tEN DC CtiKJ=SUBSlR(Sl'l(MZJ,IQ,J-lQJ ; IK=IK+l ; IQ=O+l ; ENO ; END ; IK=IK-1 ; IF IK>l THEN DO ; ST: DO P=2 BY 1 TO IK ; IF C(P-ll>ClPl THEN GL TG DO ; GO TO EN ; 00: TEMP=C ( P) ; C ( P) =C ( P-1) ; C(P-U=TE~P ; GO TO ST ; EN: END ; END ; ST lC MZ) =1 ' ; DO P=l BY l TO !K ; s"f 1( Mz ) = sll( Ml ) I I c ( p ) I I ' ' ' ; END ; END ; FREE C ; I* CHECK FOR THE SAME ELEMENTS *I 00 P=l BY 1 TO M-1 ; DO O=P+l BY 1 TC M ; IF STl(P)=Sll(O) & NUEltPJ=NUEl(O) THEN DO NUl(P)=NUl(Pl+NUl(O) ; NUl C0) =0 ; END ; END ; END ; 00 P=l BY 1 TO M-1 ; IF NUlC P)=O THE-N G!l TO ZAZ ; XYZ=•t•ltNUl(PliiS"fl(P); 00 O=P+l BY 1 TO M ; IF NU1{0)=0 THFN GO TU OF ; IF NUE1(0)=99 THEN GO TO QE ; IF NUEl(P)=NUEl(O) THEN 00 ; ' NU El ( 0) ,:9q ENO ; QE: END ; Sll(P)=XYZI l 1 l'l I 1 E1 1 I~Ufl(P) ; ZAZ: END ; IF NlJlC M)=O THEN GO T:J SZ ; IF NUE1CMJ=99 lHEN GO TO Sf ; s l u M ) = I ( I I I NU 1( M ) I I s T 1( ~ ) I I ) E I I I•.J u E 1 (M ) SZ: DO P= 2 BY 1 TO f-1 ; IF NUEl(P-l)>NUEl(P) THEN GO TO Ol ; GO TO El ; oz: ~f~r;t!lf~~P!lJE~F~~~t~~!N~tl~~\iNVE~~~~CP)=NUElfP-1) ; STlCP-l)=TEMP ~ NUlCP-l)=TEMF ; NUEl(P-lJ=TEML ; GO Tu SZ ; EZ: END : PUT PAGE LI~l I'ANSWEk =•) ; PUT ~KlPC2) ; I* LIST FINAL ELEMENT *I IX=O ; Du 0=1 BY 1 TO M ; IF NUE1(0J=99 THEN GO TO EA ; lF NU1(0)=0 THEN GO TO EA ; IF STl(O)='' THEN GO TO [A ; F=lNDEXCST1CO),• ') ; 1)0 WHILE (F>O) ; STlfOJ=SUBSTRCSll(O),l,F-lJII SUHSTR(ST1(0) 1 F+l,LENGTHCSTl(G)J-FJ ; F=INDEX(STl(O),• 'J ; END ; F=INOEXCSllC0) 9 1 ,) 1 ); IF F>O THEN s T lC 0 , =su 8 ~ T R( s 1 1 ( tJ ) t l t F- u I I SUBSTRCSTlCOJ,F+l,LENGTHfSTl(O))-FJ ; IX=IX+l • IF IX>l fHEN SllC0)= 1 +'11STUO) ; PUT SKIP LIST CST110)) ; EA: END ; IF IX=O THEN PUT SKIP Li~T ( 1 0 1 ) ; FREE STl ; FREE NUl ; FREE NUE 1 ; GO TO STA ; OUTT: END ;